How to integrate a three products












12












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I tried to integrate $x e^x sin x$, using integration by parts, and setting $dv/dx = e^x sin x$. Even though I got really close, I kept getting it wrong. Can someone please solve it with working out? Thanks in advance.



***EDIT**** I have found the answer, all thanks to those who contributed :) I don't know if I could of connected the answer here but I just posted it below, thank you all again!










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  • $begingroup$
    this aint that straight forward.
    $endgroup$
    – Lost1
    Jan 13 '14 at 11:07










  • $begingroup$
    Im not sure if you can read my writing but i think i got pretty close to the answer "wolframalpha", @Lost1. fluorine-silver.blogspot.com.au/2014/01/math-so-damn-close.html
    $endgroup$
    – Samir Chahine
    Jan 13 '14 at 11:11










  • $begingroup$
    i dont understand your v and v'. this question is such a pain in the arse if you don't use the hint Nigel posted, but needs complex numbers. I wanted to post a solution using elementary integration and i got stuck...
    $endgroup$
    – Lost1
    Jan 13 '14 at 11:14










  • $begingroup$
    Related : math.stackexchange.com/questions/540295/integrate-eax-sinbx
    $endgroup$
    – lab bhattacharjee
    Jan 13 '14 at 11:15






  • 2




    $begingroup$
    There is no such thing as a "three product". The function $xmapsto xmathrm e^xsin x$ is a function, dot. You could write it as well $2sqrt xleft(mathrm e^{x/2}right)^2sqrt xsin(x/2)cos(x/2)$ and it would be a six function...
    $endgroup$
    – Tom-Tom
    Jan 13 '14 at 16:53


















12












$begingroup$


I tried to integrate $x e^x sin x$, using integration by parts, and setting $dv/dx = e^x sin x$. Even though I got really close, I kept getting it wrong. Can someone please solve it with working out? Thanks in advance.



***EDIT**** I have found the answer, all thanks to those who contributed :) I don't know if I could of connected the answer here but I just posted it below, thank you all again!










share|cite|improve this question











$endgroup$












  • $begingroup$
    this aint that straight forward.
    $endgroup$
    – Lost1
    Jan 13 '14 at 11:07










  • $begingroup$
    Im not sure if you can read my writing but i think i got pretty close to the answer "wolframalpha", @Lost1. fluorine-silver.blogspot.com.au/2014/01/math-so-damn-close.html
    $endgroup$
    – Samir Chahine
    Jan 13 '14 at 11:11










  • $begingroup$
    i dont understand your v and v'. this question is such a pain in the arse if you don't use the hint Nigel posted, but needs complex numbers. I wanted to post a solution using elementary integration and i got stuck...
    $endgroup$
    – Lost1
    Jan 13 '14 at 11:14










  • $begingroup$
    Related : math.stackexchange.com/questions/540295/integrate-eax-sinbx
    $endgroup$
    – lab bhattacharjee
    Jan 13 '14 at 11:15






  • 2




    $begingroup$
    There is no such thing as a "three product". The function $xmapsto xmathrm e^xsin x$ is a function, dot. You could write it as well $2sqrt xleft(mathrm e^{x/2}right)^2sqrt xsin(x/2)cos(x/2)$ and it would be a six function...
    $endgroup$
    – Tom-Tom
    Jan 13 '14 at 16:53
















12












12








12


4



$begingroup$


I tried to integrate $x e^x sin x$, using integration by parts, and setting $dv/dx = e^x sin x$. Even though I got really close, I kept getting it wrong. Can someone please solve it with working out? Thanks in advance.



***EDIT**** I have found the answer, all thanks to those who contributed :) I don't know if I could of connected the answer here but I just posted it below, thank you all again!










share|cite|improve this question











$endgroup$




I tried to integrate $x e^x sin x$, using integration by parts, and setting $dv/dx = e^x sin x$. Even though I got really close, I kept getting it wrong. Can someone please solve it with working out? Thanks in advance.



***EDIT**** I have found the answer, all thanks to those who contributed :) I don't know if I could of connected the answer here but I just posted it below, thank you all again!







integration indefinite-integrals






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 13 '14 at 13:48







Samir Chahine

















asked Jan 13 '14 at 11:06









Samir ChahineSamir Chahine

6911622




6911622












  • $begingroup$
    this aint that straight forward.
    $endgroup$
    – Lost1
    Jan 13 '14 at 11:07










  • $begingroup$
    Im not sure if you can read my writing but i think i got pretty close to the answer "wolframalpha", @Lost1. fluorine-silver.blogspot.com.au/2014/01/math-so-damn-close.html
    $endgroup$
    – Samir Chahine
    Jan 13 '14 at 11:11










  • $begingroup$
    i dont understand your v and v'. this question is such a pain in the arse if you don't use the hint Nigel posted, but needs complex numbers. I wanted to post a solution using elementary integration and i got stuck...
    $endgroup$
    – Lost1
    Jan 13 '14 at 11:14










  • $begingroup$
    Related : math.stackexchange.com/questions/540295/integrate-eax-sinbx
    $endgroup$
    – lab bhattacharjee
    Jan 13 '14 at 11:15






  • 2




    $begingroup$
    There is no such thing as a "three product". The function $xmapsto xmathrm e^xsin x$ is a function, dot. You could write it as well $2sqrt xleft(mathrm e^{x/2}right)^2sqrt xsin(x/2)cos(x/2)$ and it would be a six function...
    $endgroup$
    – Tom-Tom
    Jan 13 '14 at 16:53




















  • $begingroup$
    this aint that straight forward.
    $endgroup$
    – Lost1
    Jan 13 '14 at 11:07










  • $begingroup$
    Im not sure if you can read my writing but i think i got pretty close to the answer "wolframalpha", @Lost1. fluorine-silver.blogspot.com.au/2014/01/math-so-damn-close.html
    $endgroup$
    – Samir Chahine
    Jan 13 '14 at 11:11










  • $begingroup$
    i dont understand your v and v'. this question is such a pain in the arse if you don't use the hint Nigel posted, but needs complex numbers. I wanted to post a solution using elementary integration and i got stuck...
    $endgroup$
    – Lost1
    Jan 13 '14 at 11:14










  • $begingroup$
    Related : math.stackexchange.com/questions/540295/integrate-eax-sinbx
    $endgroup$
    – lab bhattacharjee
    Jan 13 '14 at 11:15






  • 2




    $begingroup$
    There is no such thing as a "three product". The function $xmapsto xmathrm e^xsin x$ is a function, dot. You could write it as well $2sqrt xleft(mathrm e^{x/2}right)^2sqrt xsin(x/2)cos(x/2)$ and it would be a six function...
    $endgroup$
    – Tom-Tom
    Jan 13 '14 at 16:53


















$begingroup$
this aint that straight forward.
$endgroup$
– Lost1
Jan 13 '14 at 11:07




$begingroup$
this aint that straight forward.
$endgroup$
– Lost1
Jan 13 '14 at 11:07












$begingroup$
Im not sure if you can read my writing but i think i got pretty close to the answer "wolframalpha", @Lost1. fluorine-silver.blogspot.com.au/2014/01/math-so-damn-close.html
$endgroup$
– Samir Chahine
Jan 13 '14 at 11:11




$begingroup$
Im not sure if you can read my writing but i think i got pretty close to the answer "wolframalpha", @Lost1. fluorine-silver.blogspot.com.au/2014/01/math-so-damn-close.html
$endgroup$
– Samir Chahine
Jan 13 '14 at 11:11












$begingroup$
i dont understand your v and v'. this question is such a pain in the arse if you don't use the hint Nigel posted, but needs complex numbers. I wanted to post a solution using elementary integration and i got stuck...
$endgroup$
– Lost1
Jan 13 '14 at 11:14




$begingroup$
i dont understand your v and v'. this question is such a pain in the arse if you don't use the hint Nigel posted, but needs complex numbers. I wanted to post a solution using elementary integration and i got stuck...
$endgroup$
– Lost1
Jan 13 '14 at 11:14












$begingroup$
Related : math.stackexchange.com/questions/540295/integrate-eax-sinbx
$endgroup$
– lab bhattacharjee
Jan 13 '14 at 11:15




$begingroup$
Related : math.stackexchange.com/questions/540295/integrate-eax-sinbx
$endgroup$
– lab bhattacharjee
Jan 13 '14 at 11:15




2




2




$begingroup$
There is no such thing as a "three product". The function $xmapsto xmathrm e^xsin x$ is a function, dot. You could write it as well $2sqrt xleft(mathrm e^{x/2}right)^2sqrt xsin(x/2)cos(x/2)$ and it would be a six function...
$endgroup$
– Tom-Tom
Jan 13 '14 at 16:53






$begingroup$
There is no such thing as a "three product". The function $xmapsto xmathrm e^xsin x$ is a function, dot. You could write it as well $2sqrt xleft(mathrm e^{x/2}right)^2sqrt xsin(x/2)cos(x/2)$ and it would be a six function...
$endgroup$
– Tom-Tom
Jan 13 '14 at 16:53












6 Answers
6






active

oldest

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14












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If you don't want to try with complex calculus, you might use this:



Let's define $g(x) = e^x sin{x} $, so we have: $J = int x g(x) , dx$.



Then, if you use chain rule, you will have:



$$J = x, int g(x) , dx - iint g(x) , dx^2, $$



where, using again the chain rule:



$$int g(x) , dx = int e^x sin{x} , dx = frac{e^x}{2} (sin{x} - cos{x})$$



Thus:



$$iint g(x) dx^2 = int frac{e^x}{2} (sin{x} - cos{x}) , dx $$



where the first integral has already been computed. Using again the chain rule for the cosine integral, it finally yields:




$$large{
color{blue}{
J = frac{x}{2} e^x (sin{x} - cos{x} ) + frac{e^x}{2} cos{x}
}
}$$




Do not forget the integration constant! Cheers.






share|cite|improve this answer











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  • 3




    $begingroup$
    yes this is nice +1
    $endgroup$
    – Lost1
    Jan 13 '14 at 11:26












  • $begingroup$
    Sorry, I made a mistake on the transcription.
    $endgroup$
    – Dmoreno
    Jan 13 '14 at 11:32





















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One general idea with products of three functions is to use the product rule in the form
$$
(u v w)' = u' v w + u v' w + uv w'
$$
and the get partial integration in the form
$$
int u' v w = uvw - int u v' w - int uv w'
$$
and then the solution of your problem is straightforward but tedious.



After two applications of above rule (with $u=e^x$) and some reorganization you find

$$
2 int x e^x sin x , dx = xe^x sin x - x e^x cos x -int e^x sin x , dx + int e^x cos x , dx
$$
and the rest is easy.






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    7












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    Let me elaborate on Nigel's hint, and btw he meant $e^{ix} = i sin x +cos x$. There is no $pi$.



    then the integral you want is J. define the integral $I = int x cos x e^x text{d}x$.



    Then $I + iJ = int x e^{ix}e^x text{d}x = int xe^{(i+1)x}text{d}x$



    note $i$ is square root of minus one, so it i just a constant, you integrate this by part and seperate out real and imaginary part. The imaginary part is what you want.



    At first I wanted to post this, but I guessed this is not what OP wanted.



    however, without this, this integral is a pain in the backside...






    share|cite|improve this answer











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    • $begingroup$
      @Ruslan lol....
      $endgroup$
      – Lost1
      Jan 13 '14 at 11:20










    • $begingroup$
      Guess I have to learn complex numbers, thank you, if you do find out a way to solve this without it please do post it :)
      $endgroup$
      – Samir Chahine
      Jan 13 '14 at 11:20










    • $begingroup$
      @SamirChahine there is, i am sure you can crack it if you give it another 3 hours, using integration by parts...
      $endgroup$
      – Lost1
      Jan 13 '14 at 11:21










    • $begingroup$
      I will do it, and when I do I will post it, I shall never give up. @Lost1, may I ask how long it would take to learn complex numbers? And do I need to know any other type of math before I dive in
      $endgroup$
      – Samir Chahine
      Jan 13 '14 at 11:21












    • $begingroup$
      @SamirChahine take $du/dx = x, v = e^x sin x$ as your first step, and work from there
      $endgroup$
      – Lost1
      Jan 13 '14 at 11:22



















    5












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    Hint, using complex numbers:
    $$ sin(theta) = frac{ e^{itheta} - e^{-i theta} }{2i} $$



    Without complex numbers:
    Let $f = x$, $g' = e^x sin x$.



    First we calculate $g = int g' dx$ by integration by parts:
    $$ begin{array}{c}
    I = int {{e^{ax}}sin (bx)dx} = left[ {begin{array}{*{20}{c}}
    {u = sin (bx)}&{v' = {e^{ax}}}\
    {u' = bcos (bx)}&{v = {e^{ax}}/a}
    end{array}} right]mathop = limits^{{mathop{rm int}} } frac{{{e^{ax}}sin (bx)}}{a} - int {bcos (bx)frac{{{e^{ax}}}}{a}dx} \
    = frac{{{e^{ax}}sin (bx)}}{a} - frac{b}{a}int {cos (bx){e^{ax}}dx} = \
    = left[ {begin{array}{*{20}{c}}
    {f = cos (bx)}&{g' = {e^{ax}}}\
    {f' = - bsin (bx)}&{g = {e^{ax}}/a}
    end{array}} right] = frac{{{e^{ax}}sin (bx)}}{a} - frac{b}{a}left( {frac{{cos (bx){e^{ax}}}}{a} - frac{{ - b}}{a}int {sin (bx){e^{ax}}dx} } right) = \
    = frac{{{e^{ax}}sin (bx)}}{a} - frac{{bcos (bx){e^{ax}}}}{{{a^2}}} - frac{{{b^2}}}{{{a^2}}}int {sin (bx){e^{ax}}dx} = frac{{{e^{ax}}sin (bx)}}{a} - frac{{bcos (bx){e^{ax}}}}{{{a^2}}} - frac{{{b^2}}}{{{a^2}}}I
    end{array}$$

    Thus
    $$ I = frac{{{e^{ax}}sin (bx)}}{a} - frac{{bcos (bx){e^{ax}}}}{{{a^2}}} - frac{{{b^2}}}{{{a^2}}}I$$
    Solving for $I$, we get
    $$ I = frac{{{e^{ax}}left( {asin (bx) - bcos (bx)} right)}}{{{a^2} + {b^2}}} + C $$
    so
    $$ g(x) = frac{{{e^{x}}left( {sin (x) -cos (x)} right)}}{{2}} . $$



    Thus, to solve the big integral we do again integration by parts with $f=x$:
    $$ int f g' = fg - int f' g = x frac{{{e^{x}}left( {sin (x) -cos (x)} right)}}{{2}} - int left( frac{{{e^{x}}left( {sin (x) -cos (x)} right)}}{{2}} right) dx $$
    where the last integral can be calculated as above.






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      Nice double differentiation of $sin$ :)
      $endgroup$
      – Ruslan
      Jan 13 '14 at 11:23



















    4












    $begingroup$

    Hint:
    $$sin(theta)= Im (e^{i theta})$$
    Then use integration by parts.






    share|cite|improve this answer











    $endgroup$









    • 1




      $begingroup$
      I have absolutely no idea what that is, a hint for this hint please?
      $endgroup$
      – Samir Chahine
      Jan 13 '14 at 11:12






    • 1




      $begingroup$
      Hint for the hint: use Euler formula.
      $endgroup$
      – Ruslan
      Jan 13 '14 at 11:15










    • $begingroup$
      You're absolutely right. Thanks and fixed it.
      $endgroup$
      – Nigel Overmars
      Jan 13 '14 at 11:21



















    3












    $begingroup$

    $$int e^xsin(x) dx = frac{1}{2}e^x(sin x-cos x) + C
    $$



    $$
    int e^xcos(x) dx = frac{1}{2}e^x(sin x +cos x ) + C
    $$



    $$
    u = x \ u'= 1\\v'=e^xsin(x) \v= frac{1}{2}e^x(sin x -cos x) + C
    \
    I = frac{xe^x}{2}(sin x-cos x )-frac{1}{2} int e^x(sin x-cos x) + C
    \
    text{Let }Z^- = sin x -cos x
    \
    text{and}\
    text{Let }Z^+ = sin x+cos x
    \
    frac {xe^x}{2}(Z^-) - frac{1}{2}left(frac{1}{2}e^x(Z^- - Z^+)right) +C\
    frac {xe^x}{2}(sin x-cos x) - frac{1}{2}left(frac{1}{2}e^x(-2cos x )right)+C\
    frac {xe^x}{2}(sin x-cos x) + frac{1}{2}e^x(cos x) + C\
    int xe^xcos x dx = frac{1}{2}e^x(xsin x-xcos x+cos x) + C
    $$
    Well. I told you guys I would do it, thank you all for your help, turn out I've been stuck on this bastard for 3 days not two, time flies when you're having fun! ;) - I entered most of the steps, you guys should see the gaps if there are any, and i'm sorry my MathJax isn't perfect, this is my first time using it for such a big equation, thank you all again! Next stop, five function ;) (I checked on wolframalpha.com, got the same answer!)






    share|cite|improve this answer











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      6 Answers
      6






      active

      oldest

      votes








      6 Answers
      6






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      14












      $begingroup$

      If you don't want to try with complex calculus, you might use this:



      Let's define $g(x) = e^x sin{x} $, so we have: $J = int x g(x) , dx$.



      Then, if you use chain rule, you will have:



      $$J = x, int g(x) , dx - iint g(x) , dx^2, $$



      where, using again the chain rule:



      $$int g(x) , dx = int e^x sin{x} , dx = frac{e^x}{2} (sin{x} - cos{x})$$



      Thus:



      $$iint g(x) dx^2 = int frac{e^x}{2} (sin{x} - cos{x}) , dx $$



      where the first integral has already been computed. Using again the chain rule for the cosine integral, it finally yields:




      $$large{
      color{blue}{
      J = frac{x}{2} e^x (sin{x} - cos{x} ) + frac{e^x}{2} cos{x}
      }
      }$$




      Do not forget the integration constant! Cheers.






      share|cite|improve this answer











      $endgroup$









      • 3




        $begingroup$
        yes this is nice +1
        $endgroup$
        – Lost1
        Jan 13 '14 at 11:26












      • $begingroup$
        Sorry, I made a mistake on the transcription.
        $endgroup$
        – Dmoreno
        Jan 13 '14 at 11:32


















      14












      $begingroup$

      If you don't want to try with complex calculus, you might use this:



      Let's define $g(x) = e^x sin{x} $, so we have: $J = int x g(x) , dx$.



      Then, if you use chain rule, you will have:



      $$J = x, int g(x) , dx - iint g(x) , dx^2, $$



      where, using again the chain rule:



      $$int g(x) , dx = int e^x sin{x} , dx = frac{e^x}{2} (sin{x} - cos{x})$$



      Thus:



      $$iint g(x) dx^2 = int frac{e^x}{2} (sin{x} - cos{x}) , dx $$



      where the first integral has already been computed. Using again the chain rule for the cosine integral, it finally yields:




      $$large{
      color{blue}{
      J = frac{x}{2} e^x (sin{x} - cos{x} ) + frac{e^x}{2} cos{x}
      }
      }$$




      Do not forget the integration constant! Cheers.






      share|cite|improve this answer











      $endgroup$









      • 3




        $begingroup$
        yes this is nice +1
        $endgroup$
        – Lost1
        Jan 13 '14 at 11:26












      • $begingroup$
        Sorry, I made a mistake on the transcription.
        $endgroup$
        – Dmoreno
        Jan 13 '14 at 11:32
















      14












      14








      14





      $begingroup$

      If you don't want to try with complex calculus, you might use this:



      Let's define $g(x) = e^x sin{x} $, so we have: $J = int x g(x) , dx$.



      Then, if you use chain rule, you will have:



      $$J = x, int g(x) , dx - iint g(x) , dx^2, $$



      where, using again the chain rule:



      $$int g(x) , dx = int e^x sin{x} , dx = frac{e^x}{2} (sin{x} - cos{x})$$



      Thus:



      $$iint g(x) dx^2 = int frac{e^x}{2} (sin{x} - cos{x}) , dx $$



      where the first integral has already been computed. Using again the chain rule for the cosine integral, it finally yields:




      $$large{
      color{blue}{
      J = frac{x}{2} e^x (sin{x} - cos{x} ) + frac{e^x}{2} cos{x}
      }
      }$$




      Do not forget the integration constant! Cheers.






      share|cite|improve this answer











      $endgroup$



      If you don't want to try with complex calculus, you might use this:



      Let's define $g(x) = e^x sin{x} $, so we have: $J = int x g(x) , dx$.



      Then, if you use chain rule, you will have:



      $$J = x, int g(x) , dx - iint g(x) , dx^2, $$



      where, using again the chain rule:



      $$int g(x) , dx = int e^x sin{x} , dx = frac{e^x}{2} (sin{x} - cos{x})$$



      Thus:



      $$iint g(x) dx^2 = int frac{e^x}{2} (sin{x} - cos{x}) , dx $$



      where the first integral has already been computed. Using again the chain rule for the cosine integral, it finally yields:




      $$large{
      color{blue}{
      J = frac{x}{2} e^x (sin{x} - cos{x} ) + frac{e^x}{2} cos{x}
      }
      }$$




      Do not forget the integration constant! Cheers.







      share|cite|improve this answer














      share|cite|improve this answer



      share|cite|improve this answer








      edited Jun 5 '17 at 23:10

























      answered Jan 13 '14 at 11:23









      DmorenoDmoreno

      6,53131240




      6,53131240








      • 3




        $begingroup$
        yes this is nice +1
        $endgroup$
        – Lost1
        Jan 13 '14 at 11:26












      • $begingroup$
        Sorry, I made a mistake on the transcription.
        $endgroup$
        – Dmoreno
        Jan 13 '14 at 11:32
















      • 3




        $begingroup$
        yes this is nice +1
        $endgroup$
        – Lost1
        Jan 13 '14 at 11:26












      • $begingroup$
        Sorry, I made a mistake on the transcription.
        $endgroup$
        – Dmoreno
        Jan 13 '14 at 11:32










      3




      3




      $begingroup$
      yes this is nice +1
      $endgroup$
      – Lost1
      Jan 13 '14 at 11:26






      $begingroup$
      yes this is nice +1
      $endgroup$
      – Lost1
      Jan 13 '14 at 11:26














      $begingroup$
      Sorry, I made a mistake on the transcription.
      $endgroup$
      – Dmoreno
      Jan 13 '14 at 11:32






      $begingroup$
      Sorry, I made a mistake on the transcription.
      $endgroup$
      – Dmoreno
      Jan 13 '14 at 11:32













      8












      $begingroup$

      One general idea with products of three functions is to use the product rule in the form
      $$
      (u v w)' = u' v w + u v' w + uv w'
      $$
      and the get partial integration in the form
      $$
      int u' v w = uvw - int u v' w - int uv w'
      $$
      and then the solution of your problem is straightforward but tedious.



      After two applications of above rule (with $u=e^x$) and some reorganization you find

      $$
      2 int x e^x sin x , dx = xe^x sin x - x e^x cos x -int e^x sin x , dx + int e^x cos x , dx
      $$
      and the rest is easy.






      share|cite|improve this answer









      $endgroup$


















        8












        $begingroup$

        One general idea with products of three functions is to use the product rule in the form
        $$
        (u v w)' = u' v w + u v' w + uv w'
        $$
        and the get partial integration in the form
        $$
        int u' v w = uvw - int u v' w - int uv w'
        $$
        and then the solution of your problem is straightforward but tedious.



        After two applications of above rule (with $u=e^x$) and some reorganization you find

        $$
        2 int x e^x sin x , dx = xe^x sin x - x e^x cos x -int e^x sin x , dx + int e^x cos x , dx
        $$
        and the rest is easy.






        share|cite|improve this answer









        $endgroup$
















          8












          8








          8





          $begingroup$

          One general idea with products of three functions is to use the product rule in the form
          $$
          (u v w)' = u' v w + u v' w + uv w'
          $$
          and the get partial integration in the form
          $$
          int u' v w = uvw - int u v' w - int uv w'
          $$
          and then the solution of your problem is straightforward but tedious.



          After two applications of above rule (with $u=e^x$) and some reorganization you find

          $$
          2 int x e^x sin x , dx = xe^x sin x - x e^x cos x -int e^x sin x , dx + int e^x cos x , dx
          $$
          and the rest is easy.






          share|cite|improve this answer









          $endgroup$



          One general idea with products of three functions is to use the product rule in the form
          $$
          (u v w)' = u' v w + u v' w + uv w'
          $$
          and the get partial integration in the form
          $$
          int u' v w = uvw - int u v' w - int uv w'
          $$
          and then the solution of your problem is straightforward but tedious.



          After two applications of above rule (with $u=e^x$) and some reorganization you find

          $$
          2 int x e^x sin x , dx = xe^x sin x - x e^x cos x -int e^x sin x , dx + int e^x cos x , dx
          $$
          and the rest is easy.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 13 '14 at 11:59









          kjetil b halvorsenkjetil b halvorsen

          4,75742638




          4,75742638























              7












              $begingroup$

              Let me elaborate on Nigel's hint, and btw he meant $e^{ix} = i sin x +cos x$. There is no $pi$.



              then the integral you want is J. define the integral $I = int x cos x e^x text{d}x$.



              Then $I + iJ = int x e^{ix}e^x text{d}x = int xe^{(i+1)x}text{d}x$



              note $i$ is square root of minus one, so it i just a constant, you integrate this by part and seperate out real and imaginary part. The imaginary part is what you want.



              At first I wanted to post this, but I guessed this is not what OP wanted.



              however, without this, this integral is a pain in the backside...






              share|cite|improve this answer











              $endgroup$













              • $begingroup$
                @Ruslan lol....
                $endgroup$
                – Lost1
                Jan 13 '14 at 11:20










              • $begingroup$
                Guess I have to learn complex numbers, thank you, if you do find out a way to solve this without it please do post it :)
                $endgroup$
                – Samir Chahine
                Jan 13 '14 at 11:20










              • $begingroup$
                @SamirChahine there is, i am sure you can crack it if you give it another 3 hours, using integration by parts...
                $endgroup$
                – Lost1
                Jan 13 '14 at 11:21










              • $begingroup$
                I will do it, and when I do I will post it, I shall never give up. @Lost1, may I ask how long it would take to learn complex numbers? And do I need to know any other type of math before I dive in
                $endgroup$
                – Samir Chahine
                Jan 13 '14 at 11:21












              • $begingroup$
                @SamirChahine take $du/dx = x, v = e^x sin x$ as your first step, and work from there
                $endgroup$
                – Lost1
                Jan 13 '14 at 11:22
















              7












              $begingroup$

              Let me elaborate on Nigel's hint, and btw he meant $e^{ix} = i sin x +cos x$. There is no $pi$.



              then the integral you want is J. define the integral $I = int x cos x e^x text{d}x$.



              Then $I + iJ = int x e^{ix}e^x text{d}x = int xe^{(i+1)x}text{d}x$



              note $i$ is square root of minus one, so it i just a constant, you integrate this by part and seperate out real and imaginary part. The imaginary part is what you want.



              At first I wanted to post this, but I guessed this is not what OP wanted.



              however, without this, this integral is a pain in the backside...






              share|cite|improve this answer











              $endgroup$













              • $begingroup$
                @Ruslan lol....
                $endgroup$
                – Lost1
                Jan 13 '14 at 11:20










              • $begingroup$
                Guess I have to learn complex numbers, thank you, if you do find out a way to solve this without it please do post it :)
                $endgroup$
                – Samir Chahine
                Jan 13 '14 at 11:20










              • $begingroup$
                @SamirChahine there is, i am sure you can crack it if you give it another 3 hours, using integration by parts...
                $endgroup$
                – Lost1
                Jan 13 '14 at 11:21










              • $begingroup$
                I will do it, and when I do I will post it, I shall never give up. @Lost1, may I ask how long it would take to learn complex numbers? And do I need to know any other type of math before I dive in
                $endgroup$
                – Samir Chahine
                Jan 13 '14 at 11:21












              • $begingroup$
                @SamirChahine take $du/dx = x, v = e^x sin x$ as your first step, and work from there
                $endgroup$
                – Lost1
                Jan 13 '14 at 11:22














              7












              7








              7





              $begingroup$

              Let me elaborate on Nigel's hint, and btw he meant $e^{ix} = i sin x +cos x$. There is no $pi$.



              then the integral you want is J. define the integral $I = int x cos x e^x text{d}x$.



              Then $I + iJ = int x e^{ix}e^x text{d}x = int xe^{(i+1)x}text{d}x$



              note $i$ is square root of minus one, so it i just a constant, you integrate this by part and seperate out real and imaginary part. The imaginary part is what you want.



              At first I wanted to post this, but I guessed this is not what OP wanted.



              however, without this, this integral is a pain in the backside...






              share|cite|improve this answer











              $endgroup$



              Let me elaborate on Nigel's hint, and btw he meant $e^{ix} = i sin x +cos x$. There is no $pi$.



              then the integral you want is J. define the integral $I = int x cos x e^x text{d}x$.



              Then $I + iJ = int x e^{ix}e^x text{d}x = int xe^{(i+1)x}text{d}x$



              note $i$ is square root of minus one, so it i just a constant, you integrate this by part and seperate out real and imaginary part. The imaginary part is what you want.



              At first I wanted to post this, but I guessed this is not what OP wanted.



              however, without this, this integral is a pain in the backside...







              share|cite|improve this answer














              share|cite|improve this answer



              share|cite|improve this answer








              edited Jan 13 '14 at 11:23

























              answered Jan 13 '14 at 11:18









              Lost1Lost1

              5,56933370




              5,56933370












              • $begingroup$
                @Ruslan lol....
                $endgroup$
                – Lost1
                Jan 13 '14 at 11:20










              • $begingroup$
                Guess I have to learn complex numbers, thank you, if you do find out a way to solve this without it please do post it :)
                $endgroup$
                – Samir Chahine
                Jan 13 '14 at 11:20










              • $begingroup$
                @SamirChahine there is, i am sure you can crack it if you give it another 3 hours, using integration by parts...
                $endgroup$
                – Lost1
                Jan 13 '14 at 11:21










              • $begingroup$
                I will do it, and when I do I will post it, I shall never give up. @Lost1, may I ask how long it would take to learn complex numbers? And do I need to know any other type of math before I dive in
                $endgroup$
                – Samir Chahine
                Jan 13 '14 at 11:21












              • $begingroup$
                @SamirChahine take $du/dx = x, v = e^x sin x$ as your first step, and work from there
                $endgroup$
                – Lost1
                Jan 13 '14 at 11:22


















              • $begingroup$
                @Ruslan lol....
                $endgroup$
                – Lost1
                Jan 13 '14 at 11:20










              • $begingroup$
                Guess I have to learn complex numbers, thank you, if you do find out a way to solve this without it please do post it :)
                $endgroup$
                – Samir Chahine
                Jan 13 '14 at 11:20










              • $begingroup$
                @SamirChahine there is, i am sure you can crack it if you give it another 3 hours, using integration by parts...
                $endgroup$
                – Lost1
                Jan 13 '14 at 11:21










              • $begingroup$
                I will do it, and when I do I will post it, I shall never give up. @Lost1, may I ask how long it would take to learn complex numbers? And do I need to know any other type of math before I dive in
                $endgroup$
                – Samir Chahine
                Jan 13 '14 at 11:21












              • $begingroup$
                @SamirChahine take $du/dx = x, v = e^x sin x$ as your first step, and work from there
                $endgroup$
                – Lost1
                Jan 13 '14 at 11:22
















              $begingroup$
              @Ruslan lol....
              $endgroup$
              – Lost1
              Jan 13 '14 at 11:20




              $begingroup$
              @Ruslan lol....
              $endgroup$
              – Lost1
              Jan 13 '14 at 11:20












              $begingroup$
              Guess I have to learn complex numbers, thank you, if you do find out a way to solve this without it please do post it :)
              $endgroup$
              – Samir Chahine
              Jan 13 '14 at 11:20




              $begingroup$
              Guess I have to learn complex numbers, thank you, if you do find out a way to solve this without it please do post it :)
              $endgroup$
              – Samir Chahine
              Jan 13 '14 at 11:20












              $begingroup$
              @SamirChahine there is, i am sure you can crack it if you give it another 3 hours, using integration by parts...
              $endgroup$
              – Lost1
              Jan 13 '14 at 11:21




              $begingroup$
              @SamirChahine there is, i am sure you can crack it if you give it another 3 hours, using integration by parts...
              $endgroup$
              – Lost1
              Jan 13 '14 at 11:21












              $begingroup$
              I will do it, and when I do I will post it, I shall never give up. @Lost1, may I ask how long it would take to learn complex numbers? And do I need to know any other type of math before I dive in
              $endgroup$
              – Samir Chahine
              Jan 13 '14 at 11:21






              $begingroup$
              I will do it, and when I do I will post it, I shall never give up. @Lost1, may I ask how long it would take to learn complex numbers? And do I need to know any other type of math before I dive in
              $endgroup$
              – Samir Chahine
              Jan 13 '14 at 11:21














              $begingroup$
              @SamirChahine take $du/dx = x, v = e^x sin x$ as your first step, and work from there
              $endgroup$
              – Lost1
              Jan 13 '14 at 11:22




              $begingroup$
              @SamirChahine take $du/dx = x, v = e^x sin x$ as your first step, and work from there
              $endgroup$
              – Lost1
              Jan 13 '14 at 11:22











              5












              $begingroup$

              Hint, using complex numbers:
              $$ sin(theta) = frac{ e^{itheta} - e^{-i theta} }{2i} $$



              Without complex numbers:
              Let $f = x$, $g' = e^x sin x$.



              First we calculate $g = int g' dx$ by integration by parts:
              $$ begin{array}{c}
              I = int {{e^{ax}}sin (bx)dx} = left[ {begin{array}{*{20}{c}}
              {u = sin (bx)}&{v' = {e^{ax}}}\
              {u' = bcos (bx)}&{v = {e^{ax}}/a}
              end{array}} right]mathop = limits^{{mathop{rm int}} } frac{{{e^{ax}}sin (bx)}}{a} - int {bcos (bx)frac{{{e^{ax}}}}{a}dx} \
              = frac{{{e^{ax}}sin (bx)}}{a} - frac{b}{a}int {cos (bx){e^{ax}}dx} = \
              = left[ {begin{array}{*{20}{c}}
              {f = cos (bx)}&{g' = {e^{ax}}}\
              {f' = - bsin (bx)}&{g = {e^{ax}}/a}
              end{array}} right] = frac{{{e^{ax}}sin (bx)}}{a} - frac{b}{a}left( {frac{{cos (bx){e^{ax}}}}{a} - frac{{ - b}}{a}int {sin (bx){e^{ax}}dx} } right) = \
              = frac{{{e^{ax}}sin (bx)}}{a} - frac{{bcos (bx){e^{ax}}}}{{{a^2}}} - frac{{{b^2}}}{{{a^2}}}int {sin (bx){e^{ax}}dx} = frac{{{e^{ax}}sin (bx)}}{a} - frac{{bcos (bx){e^{ax}}}}{{{a^2}}} - frac{{{b^2}}}{{{a^2}}}I
              end{array}$$

              Thus
              $$ I = frac{{{e^{ax}}sin (bx)}}{a} - frac{{bcos (bx){e^{ax}}}}{{{a^2}}} - frac{{{b^2}}}{{{a^2}}}I$$
              Solving for $I$, we get
              $$ I = frac{{{e^{ax}}left( {asin (bx) - bcos (bx)} right)}}{{{a^2} + {b^2}}} + C $$
              so
              $$ g(x) = frac{{{e^{x}}left( {sin (x) -cos (x)} right)}}{{2}} . $$



              Thus, to solve the big integral we do again integration by parts with $f=x$:
              $$ int f g' = fg - int f' g = x frac{{{e^{x}}left( {sin (x) -cos (x)} right)}}{{2}} - int left( frac{{{e^{x}}left( {sin (x) -cos (x)} right)}}{{2}} right) dx $$
              where the last integral can be calculated as above.






              share|cite|improve this answer











              $endgroup$













              • $begingroup$
                Nice double differentiation of $sin$ :)
                $endgroup$
                – Ruslan
                Jan 13 '14 at 11:23
















              5












              $begingroup$

              Hint, using complex numbers:
              $$ sin(theta) = frac{ e^{itheta} - e^{-i theta} }{2i} $$



              Without complex numbers:
              Let $f = x$, $g' = e^x sin x$.



              First we calculate $g = int g' dx$ by integration by parts:
              $$ begin{array}{c}
              I = int {{e^{ax}}sin (bx)dx} = left[ {begin{array}{*{20}{c}}
              {u = sin (bx)}&{v' = {e^{ax}}}\
              {u' = bcos (bx)}&{v = {e^{ax}}/a}
              end{array}} right]mathop = limits^{{mathop{rm int}} } frac{{{e^{ax}}sin (bx)}}{a} - int {bcos (bx)frac{{{e^{ax}}}}{a}dx} \
              = frac{{{e^{ax}}sin (bx)}}{a} - frac{b}{a}int {cos (bx){e^{ax}}dx} = \
              = left[ {begin{array}{*{20}{c}}
              {f = cos (bx)}&{g' = {e^{ax}}}\
              {f' = - bsin (bx)}&{g = {e^{ax}}/a}
              end{array}} right] = frac{{{e^{ax}}sin (bx)}}{a} - frac{b}{a}left( {frac{{cos (bx){e^{ax}}}}{a} - frac{{ - b}}{a}int {sin (bx){e^{ax}}dx} } right) = \
              = frac{{{e^{ax}}sin (bx)}}{a} - frac{{bcos (bx){e^{ax}}}}{{{a^2}}} - frac{{{b^2}}}{{{a^2}}}int {sin (bx){e^{ax}}dx} = frac{{{e^{ax}}sin (bx)}}{a} - frac{{bcos (bx){e^{ax}}}}{{{a^2}}} - frac{{{b^2}}}{{{a^2}}}I
              end{array}$$

              Thus
              $$ I = frac{{{e^{ax}}sin (bx)}}{a} - frac{{bcos (bx){e^{ax}}}}{{{a^2}}} - frac{{{b^2}}}{{{a^2}}}I$$
              Solving for $I$, we get
              $$ I = frac{{{e^{ax}}left( {asin (bx) - bcos (bx)} right)}}{{{a^2} + {b^2}}} + C $$
              so
              $$ g(x) = frac{{{e^{x}}left( {sin (x) -cos (x)} right)}}{{2}} . $$



              Thus, to solve the big integral we do again integration by parts with $f=x$:
              $$ int f g' = fg - int f' g = x frac{{{e^{x}}left( {sin (x) -cos (x)} right)}}{{2}} - int left( frac{{{e^{x}}left( {sin (x) -cos (x)} right)}}{{2}} right) dx $$
              where the last integral can be calculated as above.






              share|cite|improve this answer











              $endgroup$













              • $begingroup$
                Nice double differentiation of $sin$ :)
                $endgroup$
                – Ruslan
                Jan 13 '14 at 11:23














              5












              5








              5





              $begingroup$

              Hint, using complex numbers:
              $$ sin(theta) = frac{ e^{itheta} - e^{-i theta} }{2i} $$



              Without complex numbers:
              Let $f = x$, $g' = e^x sin x$.



              First we calculate $g = int g' dx$ by integration by parts:
              $$ begin{array}{c}
              I = int {{e^{ax}}sin (bx)dx} = left[ {begin{array}{*{20}{c}}
              {u = sin (bx)}&{v' = {e^{ax}}}\
              {u' = bcos (bx)}&{v = {e^{ax}}/a}
              end{array}} right]mathop = limits^{{mathop{rm int}} } frac{{{e^{ax}}sin (bx)}}{a} - int {bcos (bx)frac{{{e^{ax}}}}{a}dx} \
              = frac{{{e^{ax}}sin (bx)}}{a} - frac{b}{a}int {cos (bx){e^{ax}}dx} = \
              = left[ {begin{array}{*{20}{c}}
              {f = cos (bx)}&{g' = {e^{ax}}}\
              {f' = - bsin (bx)}&{g = {e^{ax}}/a}
              end{array}} right] = frac{{{e^{ax}}sin (bx)}}{a} - frac{b}{a}left( {frac{{cos (bx){e^{ax}}}}{a} - frac{{ - b}}{a}int {sin (bx){e^{ax}}dx} } right) = \
              = frac{{{e^{ax}}sin (bx)}}{a} - frac{{bcos (bx){e^{ax}}}}{{{a^2}}} - frac{{{b^2}}}{{{a^2}}}int {sin (bx){e^{ax}}dx} = frac{{{e^{ax}}sin (bx)}}{a} - frac{{bcos (bx){e^{ax}}}}{{{a^2}}} - frac{{{b^2}}}{{{a^2}}}I
              end{array}$$

              Thus
              $$ I = frac{{{e^{ax}}sin (bx)}}{a} - frac{{bcos (bx){e^{ax}}}}{{{a^2}}} - frac{{{b^2}}}{{{a^2}}}I$$
              Solving for $I$, we get
              $$ I = frac{{{e^{ax}}left( {asin (bx) - bcos (bx)} right)}}{{{a^2} + {b^2}}} + C $$
              so
              $$ g(x) = frac{{{e^{x}}left( {sin (x) -cos (x)} right)}}{{2}} . $$



              Thus, to solve the big integral we do again integration by parts with $f=x$:
              $$ int f g' = fg - int f' g = x frac{{{e^{x}}left( {sin (x) -cos (x)} right)}}{{2}} - int left( frac{{{e^{x}}left( {sin (x) -cos (x)} right)}}{{2}} right) dx $$
              where the last integral can be calculated as above.






              share|cite|improve this answer











              $endgroup$



              Hint, using complex numbers:
              $$ sin(theta) = frac{ e^{itheta} - e^{-i theta} }{2i} $$



              Without complex numbers:
              Let $f = x$, $g' = e^x sin x$.



              First we calculate $g = int g' dx$ by integration by parts:
              $$ begin{array}{c}
              I = int {{e^{ax}}sin (bx)dx} = left[ {begin{array}{*{20}{c}}
              {u = sin (bx)}&{v' = {e^{ax}}}\
              {u' = bcos (bx)}&{v = {e^{ax}}/a}
              end{array}} right]mathop = limits^{{mathop{rm int}} } frac{{{e^{ax}}sin (bx)}}{a} - int {bcos (bx)frac{{{e^{ax}}}}{a}dx} \
              = frac{{{e^{ax}}sin (bx)}}{a} - frac{b}{a}int {cos (bx){e^{ax}}dx} = \
              = left[ {begin{array}{*{20}{c}}
              {f = cos (bx)}&{g' = {e^{ax}}}\
              {f' = - bsin (bx)}&{g = {e^{ax}}/a}
              end{array}} right] = frac{{{e^{ax}}sin (bx)}}{a} - frac{b}{a}left( {frac{{cos (bx){e^{ax}}}}{a} - frac{{ - b}}{a}int {sin (bx){e^{ax}}dx} } right) = \
              = frac{{{e^{ax}}sin (bx)}}{a} - frac{{bcos (bx){e^{ax}}}}{{{a^2}}} - frac{{{b^2}}}{{{a^2}}}int {sin (bx){e^{ax}}dx} = frac{{{e^{ax}}sin (bx)}}{a} - frac{{bcos (bx){e^{ax}}}}{{{a^2}}} - frac{{{b^2}}}{{{a^2}}}I
              end{array}$$

              Thus
              $$ I = frac{{{e^{ax}}sin (bx)}}{a} - frac{{bcos (bx){e^{ax}}}}{{{a^2}}} - frac{{{b^2}}}{{{a^2}}}I$$
              Solving for $I$, we get
              $$ I = frac{{{e^{ax}}left( {asin (bx) - bcos (bx)} right)}}{{{a^2} + {b^2}}} + C $$
              so
              $$ g(x) = frac{{{e^{x}}left( {sin (x) -cos (x)} right)}}{{2}} . $$



              Thus, to solve the big integral we do again integration by parts with $f=x$:
              $$ int f g' = fg - int f' g = x frac{{{e^{x}}left( {sin (x) -cos (x)} right)}}{{2}} - int left( frac{{{e^{x}}left( {sin (x) -cos (x)} right)}}{{2}} right) dx $$
              where the last integral can be calculated as above.







              share|cite|improve this answer














              share|cite|improve this answer



              share|cite|improve this answer








              edited Dec 6 '18 at 15:51









              onepound

              331216




              331216










              answered Jan 13 '14 at 11:21









              LinAlgManLinAlgMan

              1,715818




              1,715818












              • $begingroup$
                Nice double differentiation of $sin$ :)
                $endgroup$
                – Ruslan
                Jan 13 '14 at 11:23


















              • $begingroup$
                Nice double differentiation of $sin$ :)
                $endgroup$
                – Ruslan
                Jan 13 '14 at 11:23
















              $begingroup$
              Nice double differentiation of $sin$ :)
              $endgroup$
              – Ruslan
              Jan 13 '14 at 11:23




              $begingroup$
              Nice double differentiation of $sin$ :)
              $endgroup$
              – Ruslan
              Jan 13 '14 at 11:23











              4












              $begingroup$

              Hint:
              $$sin(theta)= Im (e^{i theta})$$
              Then use integration by parts.






              share|cite|improve this answer











              $endgroup$









              • 1




                $begingroup$
                I have absolutely no idea what that is, a hint for this hint please?
                $endgroup$
                – Samir Chahine
                Jan 13 '14 at 11:12






              • 1




                $begingroup$
                Hint for the hint: use Euler formula.
                $endgroup$
                – Ruslan
                Jan 13 '14 at 11:15










              • $begingroup$
                You're absolutely right. Thanks and fixed it.
                $endgroup$
                – Nigel Overmars
                Jan 13 '14 at 11:21
















              4












              $begingroup$

              Hint:
              $$sin(theta)= Im (e^{i theta})$$
              Then use integration by parts.






              share|cite|improve this answer











              $endgroup$









              • 1




                $begingroup$
                I have absolutely no idea what that is, a hint for this hint please?
                $endgroup$
                – Samir Chahine
                Jan 13 '14 at 11:12






              • 1




                $begingroup$
                Hint for the hint: use Euler formula.
                $endgroup$
                – Ruslan
                Jan 13 '14 at 11:15










              • $begingroup$
                You're absolutely right. Thanks and fixed it.
                $endgroup$
                – Nigel Overmars
                Jan 13 '14 at 11:21














              4












              4








              4





              $begingroup$

              Hint:
              $$sin(theta)= Im (e^{i theta})$$
              Then use integration by parts.






              share|cite|improve this answer











              $endgroup$



              Hint:
              $$sin(theta)= Im (e^{i theta})$$
              Then use integration by parts.







              share|cite|improve this answer














              share|cite|improve this answer



              share|cite|improve this answer








              edited Jan 13 '14 at 11:21

























              answered Jan 13 '14 at 11:09









              Nigel OvermarsNigel Overmars

              2,61211225




              2,61211225








              • 1




                $begingroup$
                I have absolutely no idea what that is, a hint for this hint please?
                $endgroup$
                – Samir Chahine
                Jan 13 '14 at 11:12






              • 1




                $begingroup$
                Hint for the hint: use Euler formula.
                $endgroup$
                – Ruslan
                Jan 13 '14 at 11:15










              • $begingroup$
                You're absolutely right. Thanks and fixed it.
                $endgroup$
                – Nigel Overmars
                Jan 13 '14 at 11:21














              • 1




                $begingroup$
                I have absolutely no idea what that is, a hint for this hint please?
                $endgroup$
                – Samir Chahine
                Jan 13 '14 at 11:12






              • 1




                $begingroup$
                Hint for the hint: use Euler formula.
                $endgroup$
                – Ruslan
                Jan 13 '14 at 11:15










              • $begingroup$
                You're absolutely right. Thanks and fixed it.
                $endgroup$
                – Nigel Overmars
                Jan 13 '14 at 11:21








              1




              1




              $begingroup$
              I have absolutely no idea what that is, a hint for this hint please?
              $endgroup$
              – Samir Chahine
              Jan 13 '14 at 11:12




              $begingroup$
              I have absolutely no idea what that is, a hint for this hint please?
              $endgroup$
              – Samir Chahine
              Jan 13 '14 at 11:12




              1




              1




              $begingroup$
              Hint for the hint: use Euler formula.
              $endgroup$
              – Ruslan
              Jan 13 '14 at 11:15




              $begingroup$
              Hint for the hint: use Euler formula.
              $endgroup$
              – Ruslan
              Jan 13 '14 at 11:15












              $begingroup$
              You're absolutely right. Thanks and fixed it.
              $endgroup$
              – Nigel Overmars
              Jan 13 '14 at 11:21




              $begingroup$
              You're absolutely right. Thanks and fixed it.
              $endgroup$
              – Nigel Overmars
              Jan 13 '14 at 11:21











              3












              $begingroup$

              $$int e^xsin(x) dx = frac{1}{2}e^x(sin x-cos x) + C
              $$



              $$
              int e^xcos(x) dx = frac{1}{2}e^x(sin x +cos x ) + C
              $$



              $$
              u = x \ u'= 1\\v'=e^xsin(x) \v= frac{1}{2}e^x(sin x -cos x) + C
              \
              I = frac{xe^x}{2}(sin x-cos x )-frac{1}{2} int e^x(sin x-cos x) + C
              \
              text{Let }Z^- = sin x -cos x
              \
              text{and}\
              text{Let }Z^+ = sin x+cos x
              \
              frac {xe^x}{2}(Z^-) - frac{1}{2}left(frac{1}{2}e^x(Z^- - Z^+)right) +C\
              frac {xe^x}{2}(sin x-cos x) - frac{1}{2}left(frac{1}{2}e^x(-2cos x )right)+C\
              frac {xe^x}{2}(sin x-cos x) + frac{1}{2}e^x(cos x) + C\
              int xe^xcos x dx = frac{1}{2}e^x(xsin x-xcos x+cos x) + C
              $$
              Well. I told you guys I would do it, thank you all for your help, turn out I've been stuck on this bastard for 3 days not two, time flies when you're having fun! ;) - I entered most of the steps, you guys should see the gaps if there are any, and i'm sorry my MathJax isn't perfect, this is my first time using it for such a big equation, thank you all again! Next stop, five function ;) (I checked on wolframalpha.com, got the same answer!)






              share|cite|improve this answer











              $endgroup$


















                3












                $begingroup$

                $$int e^xsin(x) dx = frac{1}{2}e^x(sin x-cos x) + C
                $$



                $$
                int e^xcos(x) dx = frac{1}{2}e^x(sin x +cos x ) + C
                $$



                $$
                u = x \ u'= 1\\v'=e^xsin(x) \v= frac{1}{2}e^x(sin x -cos x) + C
                \
                I = frac{xe^x}{2}(sin x-cos x )-frac{1}{2} int e^x(sin x-cos x) + C
                \
                text{Let }Z^- = sin x -cos x
                \
                text{and}\
                text{Let }Z^+ = sin x+cos x
                \
                frac {xe^x}{2}(Z^-) - frac{1}{2}left(frac{1}{2}e^x(Z^- - Z^+)right) +C\
                frac {xe^x}{2}(sin x-cos x) - frac{1}{2}left(frac{1}{2}e^x(-2cos x )right)+C\
                frac {xe^x}{2}(sin x-cos x) + frac{1}{2}e^x(cos x) + C\
                int xe^xcos x dx = frac{1}{2}e^x(xsin x-xcos x+cos x) + C
                $$
                Well. I told you guys I would do it, thank you all for your help, turn out I've been stuck on this bastard for 3 days not two, time flies when you're having fun! ;) - I entered most of the steps, you guys should see the gaps if there are any, and i'm sorry my MathJax isn't perfect, this is my first time using it for such a big equation, thank you all again! Next stop, five function ;) (I checked on wolframalpha.com, got the same answer!)






                share|cite|improve this answer











                $endgroup$
















                  3












                  3








                  3





                  $begingroup$

                  $$int e^xsin(x) dx = frac{1}{2}e^x(sin x-cos x) + C
                  $$



                  $$
                  int e^xcos(x) dx = frac{1}{2}e^x(sin x +cos x ) + C
                  $$



                  $$
                  u = x \ u'= 1\\v'=e^xsin(x) \v= frac{1}{2}e^x(sin x -cos x) + C
                  \
                  I = frac{xe^x}{2}(sin x-cos x )-frac{1}{2} int e^x(sin x-cos x) + C
                  \
                  text{Let }Z^- = sin x -cos x
                  \
                  text{and}\
                  text{Let }Z^+ = sin x+cos x
                  \
                  frac {xe^x}{2}(Z^-) - frac{1}{2}left(frac{1}{2}e^x(Z^- - Z^+)right) +C\
                  frac {xe^x}{2}(sin x-cos x) - frac{1}{2}left(frac{1}{2}e^x(-2cos x )right)+C\
                  frac {xe^x}{2}(sin x-cos x) + frac{1}{2}e^x(cos x) + C\
                  int xe^xcos x dx = frac{1}{2}e^x(xsin x-xcos x+cos x) + C
                  $$
                  Well. I told you guys I would do it, thank you all for your help, turn out I've been stuck on this bastard for 3 days not two, time flies when you're having fun! ;) - I entered most of the steps, you guys should see the gaps if there are any, and i'm sorry my MathJax isn't perfect, this is my first time using it for such a big equation, thank you all again! Next stop, five function ;) (I checked on wolframalpha.com, got the same answer!)






                  share|cite|improve this answer











                  $endgroup$



                  $$int e^xsin(x) dx = frac{1}{2}e^x(sin x-cos x) + C
                  $$



                  $$
                  int e^xcos(x) dx = frac{1}{2}e^x(sin x +cos x ) + C
                  $$



                  $$
                  u = x \ u'= 1\\v'=e^xsin(x) \v= frac{1}{2}e^x(sin x -cos x) + C
                  \
                  I = frac{xe^x}{2}(sin x-cos x )-frac{1}{2} int e^x(sin x-cos x) + C
                  \
                  text{Let }Z^- = sin x -cos x
                  \
                  text{and}\
                  text{Let }Z^+ = sin x+cos x
                  \
                  frac {xe^x}{2}(Z^-) - frac{1}{2}left(frac{1}{2}e^x(Z^- - Z^+)right) +C\
                  frac {xe^x}{2}(sin x-cos x) - frac{1}{2}left(frac{1}{2}e^x(-2cos x )right)+C\
                  frac {xe^x}{2}(sin x-cos x) + frac{1}{2}e^x(cos x) + C\
                  int xe^xcos x dx = frac{1}{2}e^x(xsin x-xcos x+cos x) + C
                  $$
                  Well. I told you guys I would do it, thank you all for your help, turn out I've been stuck on this bastard for 3 days not two, time flies when you're having fun! ;) - I entered most of the steps, you guys should see the gaps if there are any, and i'm sorry my MathJax isn't perfect, this is my first time using it for such a big equation, thank you all again! Next stop, five function ;) (I checked on wolframalpha.com, got the same answer!)







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited Jan 13 '14 at 21:11









                  JMCF125

                  1,4401129




                  1,4401129










                  answered Jan 13 '14 at 13:21









                  Samir ChahineSamir Chahine

                  6911622




                  6911622






























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