Proof of exponential function integral











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I fully apologize if this is a duplicate, but I really can’t find the answer to this online here. But a proof of the rule



$$int a^x dx= frac{a^x}{ln(a)}+C$$



would be really helpful. I need it for something I’m doing. This is all I need. I have no sources on the problem as I want the proof myself, but I use this rule so much that it struck me as second nature when I wanted to solve it! Any possible answers would be appreciated.






calculus integration proof-writing exponential-function







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  • 4




    It's quite simple. $(a^x)' = a^xcdot ln(a)$, and the rule follows.
    – Jakobian
    6 hours ago










  • @Jakobian : The only real answer. +1.
    – MPW
    5 hours ago















up vote
2
down vote

favorite
1












I fully apologize if this is a duplicate, but I really can’t find the answer to this online here. But a proof of the rule



$$int a^x dx= frac{a^x}{ln(a)}+C$$



would be really helpful. I need it for something I’m doing. This is all I need. I have no sources on the problem as I want the proof myself, but I use this rule so much that it struck me as second nature when I wanted to solve it! Any possible answers would be appreciated.






calculus integration proof-writing exponential-function







share|cite|improve this question




















  • 4




    It's quite simple. $(a^x)' = a^xcdot ln(a)$, and the rule follows.
    – Jakobian
    6 hours ago










  • @Jakobian : The only real answer. +1.
    – MPW
    5 hours ago













up vote
2
down vote

favorite
1









up vote
2
down vote

favorite
1






1





I fully apologize if this is a duplicate, but I really can’t find the answer to this online here. But a proof of the rule



$$int a^x dx= frac{a^x}{ln(a)}+C$$



would be really helpful. I need it for something I’m doing. This is all I need. I have no sources on the problem as I want the proof myself, but I use this rule so much that it struck me as second nature when I wanted to solve it! Any possible answers would be appreciated.






calculus integration proof-writing exponential-function







share|cite|improve this question















I fully apologize if this is a duplicate, but I really can’t find the answer to this online here. But a proof of the rule



$$int a^x dx= frac{a^x}{ln(a)}+C$$



would be really helpful. I need it for something I’m doing. This is all I need. I have no sources on the problem as I want the proof myself, but I use this rule so much that it struck me as second nature when I wanted to solve it! Any possible answers would be appreciated.






calculus integration proof-writing exponential-function




calculus integration proof-writing exponential-function






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 5 hours ago

























asked 6 hours ago









Math Bob

138




138








  • 4




    It's quite simple. $(a^x)' = a^xcdot ln(a)$, and the rule follows.
    – Jakobian
    6 hours ago










  • @Jakobian : The only real answer. +1.
    – MPW
    5 hours ago














  • 4




    It's quite simple. $(a^x)' = a^xcdot ln(a)$, and the rule follows.
    – Jakobian
    6 hours ago










  • @Jakobian : The only real answer. +1.
    – MPW
    5 hours ago








4




4




It's quite simple. $(a^x)' = a^xcdot ln(a)$, and the rule follows.
– Jakobian
6 hours ago




It's quite simple. $(a^x)' = a^xcdot ln(a)$, and the rule follows.
– Jakobian
6 hours ago












@Jakobian : The only real answer. +1.
– MPW
5 hours ago




@Jakobian : The only real answer. +1.
– MPW
5 hours ago










3 Answers
3






active

oldest

votes

















up vote
4
down vote



accepted










I assume it's kosher to use the exponential integral with base $e$, i.e. $int e^x dx = e^x +C$? Or, more generally, for a constant $k$,



$$int e^{kx}dx = frac{1}{k}e^{kx}+C$$



If so, then note:



$$int a^x dx = int e^{ln(a^x)} dx = int e^{x ln(a)}dx = frac{1}{ln(a)}e^{x ln(a)}+C= frac{1}{ln(a)}e^{ln(a^x)}+C= frac{a^x}{ln(a)}+C$$



This is mostly just manipulation of various logarithm properties: namely,



$$e^{ln(x)} = x$$
$$ln(a^b) = b ln(a)$$



Also, a nitpick: the integral in your question, OP, needs a $+C$ after it, since indefinite integration introduces an arbitrary constant.






share|cite|improve this answer





















  • OK, so I know this already seems taken for granted, I guess (because it was used in all the other proofs I saw online), can you please furthermore prove ∫e^kx dx=1/k(e^kx)+C?
    – Math Bob
    5 hours ago












  • Personally I like proving it by using the power series definition for $e^{kx}$, i.e. $$e^{kx} = sum_{n=0}^infty frac{(kx)^n}{n!}$$ but that involves a few technicalities regarding how infinite summations work under differentiation/integration. You could always just go "well, see, $$frac{d}{dx} frac{e^{kx}}{k} + C= k frac{e^{kx}}{k} +0= e^{kx}$$ so $e^{kx}/k + C$ must be the antiderivative."
    – Eevee Trainer
    5 hours ago










  • Okay. I have taken this in so thanks for the answer!
    – Math Bob
    1 hour ago


















up vote
1
down vote













Well this one can be found within every good integration table $($e.g. take at look at this$)$. Anyway it is actually not that hard to compute therefore I will demonstrate it for you. First of all the basic property of any exponential function with a basis $a$ is that it can be rewritten in terms of $e$ in the following way



$$a^x=left(e^{ln(a)}right)^x=e^{xln(a)}$$



Now we know that $e^x$ remains $e^x$ after integration aswell as after differentiation. Adding a constant $c$ before the $x$ within the exponent yields to



$$frac d{dx}e^{cx}=ce^{cx}text{ and }int e^{cx}dx=frac1ce^{cx}+k$$



From hereon we are basically done since $ln(a)$ can be seens as a constant while integrating. So plugging this together leads to



$$int a^x dx=int e^{xln(a)}dx=frac1{ln(a)}e^{xln(a)}+k=frac{a^x}{ln(a)}+k$$




$$int a^xdx=frac{a^x}{ln(a)}+k$$







share|cite|improve this answer




























    up vote
    1
    down vote













    Just differentiate the right hand side and see what you get. Note that
    $$
    frac{d}{dx}left(frac{a^x}{ln a}right)=frac{1}{ln a}frac{d}{dx}(e^{xln a})=frac{1}{ln a}times e^{xln a}times ln a=a^x
    $$
    where we used the chain rule in the second equality.






    share|cite|improve this answer





















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      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes








      up vote
      4
      down vote



      accepted










      I assume it's kosher to use the exponential integral with base $e$, i.e. $int e^x dx = e^x +C$? Or, more generally, for a constant $k$,



      $$int e^{kx}dx = frac{1}{k}e^{kx}+C$$



      If so, then note:



      $$int a^x dx = int e^{ln(a^x)} dx = int e^{x ln(a)}dx = frac{1}{ln(a)}e^{x ln(a)}+C= frac{1}{ln(a)}e^{ln(a^x)}+C= frac{a^x}{ln(a)}+C$$



      This is mostly just manipulation of various logarithm properties: namely,



      $$e^{ln(x)} = x$$
      $$ln(a^b) = b ln(a)$$



      Also, a nitpick: the integral in your question, OP, needs a $+C$ after it, since indefinite integration introduces an arbitrary constant.






      share|cite|improve this answer





















      • OK, so I know this already seems taken for granted, I guess (because it was used in all the other proofs I saw online), can you please furthermore prove ∫e^kx dx=1/k(e^kx)+C?
        – Math Bob
        5 hours ago












      • Personally I like proving it by using the power series definition for $e^{kx}$, i.e. $$e^{kx} = sum_{n=0}^infty frac{(kx)^n}{n!}$$ but that involves a few technicalities regarding how infinite summations work under differentiation/integration. You could always just go "well, see, $$frac{d}{dx} frac{e^{kx}}{k} + C= k frac{e^{kx}}{k} +0= e^{kx}$$ so $e^{kx}/k + C$ must be the antiderivative."
        – Eevee Trainer
        5 hours ago










      • Okay. I have taken this in so thanks for the answer!
        – Math Bob
        1 hour ago















      up vote
      4
      down vote



      accepted










      I assume it's kosher to use the exponential integral with base $e$, i.e. $int e^x dx = e^x +C$? Or, more generally, for a constant $k$,



      $$int e^{kx}dx = frac{1}{k}e^{kx}+C$$



      If so, then note:



      $$int a^x dx = int e^{ln(a^x)} dx = int e^{x ln(a)}dx = frac{1}{ln(a)}e^{x ln(a)}+C= frac{1}{ln(a)}e^{ln(a^x)}+C= frac{a^x}{ln(a)}+C$$



      This is mostly just manipulation of various logarithm properties: namely,



      $$e^{ln(x)} = x$$
      $$ln(a^b) = b ln(a)$$



      Also, a nitpick: the integral in your question, OP, needs a $+C$ after it, since indefinite integration introduces an arbitrary constant.






      share|cite|improve this answer





















      • OK, so I know this already seems taken for granted, I guess (because it was used in all the other proofs I saw online), can you please furthermore prove ∫e^kx dx=1/k(e^kx)+C?
        – Math Bob
        5 hours ago












      • Personally I like proving it by using the power series definition for $e^{kx}$, i.e. $$e^{kx} = sum_{n=0}^infty frac{(kx)^n}{n!}$$ but that involves a few technicalities regarding how infinite summations work under differentiation/integration. You could always just go "well, see, $$frac{d}{dx} frac{e^{kx}}{k} + C= k frac{e^{kx}}{k} +0= e^{kx}$$ so $e^{kx}/k + C$ must be the antiderivative."
        – Eevee Trainer
        5 hours ago










      • Okay. I have taken this in so thanks for the answer!
        – Math Bob
        1 hour ago













      up vote
      4
      down vote



      accepted







      up vote
      4
      down vote



      accepted






      I assume it's kosher to use the exponential integral with base $e$, i.e. $int e^x dx = e^x +C$? Or, more generally, for a constant $k$,



      $$int e^{kx}dx = frac{1}{k}e^{kx}+C$$



      If so, then note:



      $$int a^x dx = int e^{ln(a^x)} dx = int e^{x ln(a)}dx = frac{1}{ln(a)}e^{x ln(a)}+C= frac{1}{ln(a)}e^{ln(a^x)}+C= frac{a^x}{ln(a)}+C$$



      This is mostly just manipulation of various logarithm properties: namely,



      $$e^{ln(x)} = x$$
      $$ln(a^b) = b ln(a)$$



      Also, a nitpick: the integral in your question, OP, needs a $+C$ after it, since indefinite integration introduces an arbitrary constant.






      share|cite|improve this answer












      I assume it's kosher to use the exponential integral with base $e$, i.e. $int e^x dx = e^x +C$? Or, more generally, for a constant $k$,



      $$int e^{kx}dx = frac{1}{k}e^{kx}+C$$



      If so, then note:



      $$int a^x dx = int e^{ln(a^x)} dx = int e^{x ln(a)}dx = frac{1}{ln(a)}e^{x ln(a)}+C= frac{1}{ln(a)}e^{ln(a^x)}+C= frac{a^x}{ln(a)}+C$$



      This is mostly just manipulation of various logarithm properties: namely,



      $$e^{ln(x)} = x$$
      $$ln(a^b) = b ln(a)$$



      Also, a nitpick: the integral in your question, OP, needs a $+C$ after it, since indefinite integration introduces an arbitrary constant.







      share|cite|improve this answer












      share|cite|improve this answer



      share|cite|improve this answer










      answered 6 hours ago









      Eevee Trainer

      3,320225




      3,320225












      • OK, so I know this already seems taken for granted, I guess (because it was used in all the other proofs I saw online), can you please furthermore prove ∫e^kx dx=1/k(e^kx)+C?
        – Math Bob
        5 hours ago












      • Personally I like proving it by using the power series definition for $e^{kx}$, i.e. $$e^{kx} = sum_{n=0}^infty frac{(kx)^n}{n!}$$ but that involves a few technicalities regarding how infinite summations work under differentiation/integration. You could always just go "well, see, $$frac{d}{dx} frac{e^{kx}}{k} + C= k frac{e^{kx}}{k} +0= e^{kx}$$ so $e^{kx}/k + C$ must be the antiderivative."
        – Eevee Trainer
        5 hours ago










      • Okay. I have taken this in so thanks for the answer!
        – Math Bob
        1 hour ago


















      • OK, so I know this already seems taken for granted, I guess (because it was used in all the other proofs I saw online), can you please furthermore prove ∫e^kx dx=1/k(e^kx)+C?
        – Math Bob
        5 hours ago












      • Personally I like proving it by using the power series definition for $e^{kx}$, i.e. $$e^{kx} = sum_{n=0}^infty frac{(kx)^n}{n!}$$ but that involves a few technicalities regarding how infinite summations work under differentiation/integration. You could always just go "well, see, $$frac{d}{dx} frac{e^{kx}}{k} + C= k frac{e^{kx}}{k} +0= e^{kx}$$ so $e^{kx}/k + C$ must be the antiderivative."
        – Eevee Trainer
        5 hours ago










      • Okay. I have taken this in so thanks for the answer!
        – Math Bob
        1 hour ago
















      OK, so I know this already seems taken for granted, I guess (because it was used in all the other proofs I saw online), can you please furthermore prove ∫e^kx dx=1/k(e^kx)+C?
      – Math Bob
      5 hours ago






      OK, so I know this already seems taken for granted, I guess (because it was used in all the other proofs I saw online), can you please furthermore prove ∫e^kx dx=1/k(e^kx)+C?
      – Math Bob
      5 hours ago














      Personally I like proving it by using the power series definition for $e^{kx}$, i.e. $$e^{kx} = sum_{n=0}^infty frac{(kx)^n}{n!}$$ but that involves a few technicalities regarding how infinite summations work under differentiation/integration. You could always just go "well, see, $$frac{d}{dx} frac{e^{kx}}{k} + C= k frac{e^{kx}}{k} +0= e^{kx}$$ so $e^{kx}/k + C$ must be the antiderivative."
      – Eevee Trainer
      5 hours ago




      Personally I like proving it by using the power series definition for $e^{kx}$, i.e. $$e^{kx} = sum_{n=0}^infty frac{(kx)^n}{n!}$$ but that involves a few technicalities regarding how infinite summations work under differentiation/integration. You could always just go "well, see, $$frac{d}{dx} frac{e^{kx}}{k} + C= k frac{e^{kx}}{k} +0= e^{kx}$$ so $e^{kx}/k + C$ must be the antiderivative."
      – Eevee Trainer
      5 hours ago












      Okay. I have taken this in so thanks for the answer!
      – Math Bob
      1 hour ago




      Okay. I have taken this in so thanks for the answer!
      – Math Bob
      1 hour ago










      up vote
      1
      down vote













      Well this one can be found within every good integration table $($e.g. take at look at this$)$. Anyway it is actually not that hard to compute therefore I will demonstrate it for you. First of all the basic property of any exponential function with a basis $a$ is that it can be rewritten in terms of $e$ in the following way



      $$a^x=left(e^{ln(a)}right)^x=e^{xln(a)}$$



      Now we know that $e^x$ remains $e^x$ after integration aswell as after differentiation. Adding a constant $c$ before the $x$ within the exponent yields to



      $$frac d{dx}e^{cx}=ce^{cx}text{ and }int e^{cx}dx=frac1ce^{cx}+k$$



      From hereon we are basically done since $ln(a)$ can be seens as a constant while integrating. So plugging this together leads to



      $$int a^x dx=int e^{xln(a)}dx=frac1{ln(a)}e^{xln(a)}+k=frac{a^x}{ln(a)}+k$$




      $$int a^xdx=frac{a^x}{ln(a)}+k$$







      share|cite|improve this answer

























        up vote
        1
        down vote













        Well this one can be found within every good integration table $($e.g. take at look at this$)$. Anyway it is actually not that hard to compute therefore I will demonstrate it for you. First of all the basic property of any exponential function with a basis $a$ is that it can be rewritten in terms of $e$ in the following way



        $$a^x=left(e^{ln(a)}right)^x=e^{xln(a)}$$



        Now we know that $e^x$ remains $e^x$ after integration aswell as after differentiation. Adding a constant $c$ before the $x$ within the exponent yields to



        $$frac d{dx}e^{cx}=ce^{cx}text{ and }int e^{cx}dx=frac1ce^{cx}+k$$



        From hereon we are basically done since $ln(a)$ can be seens as a constant while integrating. So plugging this together leads to



        $$int a^x dx=int e^{xln(a)}dx=frac1{ln(a)}e^{xln(a)}+k=frac{a^x}{ln(a)}+k$$




        $$int a^xdx=frac{a^x}{ln(a)}+k$$







        share|cite|improve this answer























          up vote
          1
          down vote










          up vote
          1
          down vote









          Well this one can be found within every good integration table $($e.g. take at look at this$)$. Anyway it is actually not that hard to compute therefore I will demonstrate it for you. First of all the basic property of any exponential function with a basis $a$ is that it can be rewritten in terms of $e$ in the following way



          $$a^x=left(e^{ln(a)}right)^x=e^{xln(a)}$$



          Now we know that $e^x$ remains $e^x$ after integration aswell as after differentiation. Adding a constant $c$ before the $x$ within the exponent yields to



          $$frac d{dx}e^{cx}=ce^{cx}text{ and }int e^{cx}dx=frac1ce^{cx}+k$$



          From hereon we are basically done since $ln(a)$ can be seens as a constant while integrating. So plugging this together leads to



          $$int a^x dx=int e^{xln(a)}dx=frac1{ln(a)}e^{xln(a)}+k=frac{a^x}{ln(a)}+k$$




          $$int a^xdx=frac{a^x}{ln(a)}+k$$







          share|cite|improve this answer












          Well this one can be found within every good integration table $($e.g. take at look at this$)$. Anyway it is actually not that hard to compute therefore I will demonstrate it for you. First of all the basic property of any exponential function with a basis $a$ is that it can be rewritten in terms of $e$ in the following way



          $$a^x=left(e^{ln(a)}right)^x=e^{xln(a)}$$



          Now we know that $e^x$ remains $e^x$ after integration aswell as after differentiation. Adding a constant $c$ before the $x$ within the exponent yields to



          $$frac d{dx}e^{cx}=ce^{cx}text{ and }int e^{cx}dx=frac1ce^{cx}+k$$



          From hereon we are basically done since $ln(a)$ can be seens as a constant while integrating. So plugging this together leads to



          $$int a^x dx=int e^{xln(a)}dx=frac1{ln(a)}e^{xln(a)}+k=frac{a^x}{ln(a)}+k$$




          $$int a^xdx=frac{a^x}{ln(a)}+k$$








          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 6 hours ago









          mrtaurho

          2,9651928




          2,9651928






















              up vote
              1
              down vote













              Just differentiate the right hand side and see what you get. Note that
              $$
              frac{d}{dx}left(frac{a^x}{ln a}right)=frac{1}{ln a}frac{d}{dx}(e^{xln a})=frac{1}{ln a}times e^{xln a}times ln a=a^x
              $$
              where we used the chain rule in the second equality.






              share|cite|improve this answer

























                up vote
                1
                down vote













                Just differentiate the right hand side and see what you get. Note that
                $$
                frac{d}{dx}left(frac{a^x}{ln a}right)=frac{1}{ln a}frac{d}{dx}(e^{xln a})=frac{1}{ln a}times e^{xln a}times ln a=a^x
                $$
                where we used the chain rule in the second equality.






                share|cite|improve this answer























                  up vote
                  1
                  down vote










                  up vote
                  1
                  down vote









                  Just differentiate the right hand side and see what you get. Note that
                  $$
                  frac{d}{dx}left(frac{a^x}{ln a}right)=frac{1}{ln a}frac{d}{dx}(e^{xln a})=frac{1}{ln a}times e^{xln a}times ln a=a^x
                  $$
                  where we used the chain rule in the second equality.






                  share|cite|improve this answer












                  Just differentiate the right hand side and see what you get. Note that
                  $$
                  frac{d}{dx}left(frac{a^x}{ln a}right)=frac{1}{ln a}frac{d}{dx}(e^{xln a})=frac{1}{ln a}times e^{xln a}times ln a=a^x
                  $$
                  where we used the chain rule in the second equality.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 5 hours ago









                  Foobaz John

                  20.4k41250




                  20.4k41250






























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                      Ellipse (mathématiques)

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                      Pour les articles homonymes, voir Ellipse. L'ellipse est le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante. Section du cône ou projection du cercle. En géométrie, une ellipse est une courbe plane fermée obtenue par l’intersection d’un cône de révolution avec un plan, à condition que celui-ci coupe l'axe de rotation du cône ou du cylindre : c'est une conique d'excentricité strictement comprise entre 0 et 1. On peut également la définir comme le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante (sa construction par la méthode du jardinier est très simple). Dans la vie courante, l’ellipse est la forme qu'on perçoit en regardant un cercle en perspective, ou la figure formée par l’ombre d'un disque sur une surface plane. On retrouve aussi, en première approximation [ 1 ] , des ellipses dans les trajectoires des corps célestes (planètes, comètes ou satellites arti...
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                      Quarter-circle Tiles

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                      up vote 0 down vote favorite I have $99$ identical square tiles, each with a quarter-circle drawn on it like this: [asy] size(1.5cm); draw(Arc((2,0),1,90,180),red+1); draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)); [/asy] When I arrange the tiles in a $9times 11$ rectangular grid, each with a random orientation, what is the expected value of the number of full circles I form? I think this problem has to do with finding the chance any given 2x2 square has a circle, but I can't find it. expected-value share | cite | improve this question asked Nov 20 at 15:03 6minecraftninja 1 2 ...
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                      Mont Emei

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                      Mont Emei.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Vue du sommet Wanfo du mont Emei Géographie Altitude 3 099  m , Wanfo Coordonnées 29° 31′ 11″ nord, 103° 19′ 57″ est Administration Pays   Chine Province Sichuan Ville-préfecture Leshan Géolocalisation sur la carte : Sichuan Mont Emei Géolocalisation sur la carte : Chine Mont Emei modifier   Paysage panoramique du mont Emei, incluant le paysage panoramique du grand Bouddha de Leshan *   Patrimoine mondial de l'UNESCO Passerelle en bois sur le versant ouest Pays   Chine Subdivision Sichuan Type Mixte Critères (iv) (vi) (x) Superficie 15 400 ha Numéro d’identification 779 Zone géographique Asie et Pacifique  ** Année d’inscription 1996 (20 e session) * Descriptif officiel UNESCO ** Classification géog...
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