Determine the value of (p,q)?












1














find all value of $(p,q)$ for which integral



$$int_0^1 x^pln^q(1/x)dx$$



converges ?



My answer is $p > q-1$



Is It correct?



Proof : $$int_0^1 x^pln^q(1/x)dxleint_0^1x^{p-q}dx= frac{1}{p-q+1}$$






real-analysis integration







share|cite|improve this question




















  • 4




    What about other cases? You clearly have not found all values.
    – xbh
    Nov 25 at 14:00






  • 1




    As $$int_{0}^{1}xlnleft(frac{1}{x}right)text{d}x=frac{1}{4}$$ your answer seems to be incomplete (it's only one of many examples which do not follow the condition you have found).
    – Galc127
    Nov 25 at 14:01












  • @xbh then what is the answer ?
    – jasmine
    Nov 25 at 14:06










  • @Galc127 okkss but u have take $p=q=1$
    – jasmine
    Nov 25 at 14:07






  • 1




    @jasmine, Oops, my bad... you can take $p=0,q=1$ and you have $$intlimits_{0}^{1}lnleft(frac{1}{x}right)text{d}x=1$$
    – Galc127
    Nov 25 at 14:11
















1














find all value of $(p,q)$ for which integral



$$int_0^1 x^pln^q(1/x)dx$$



converges ?



My answer is $p > q-1$



Is It correct?



Proof : $$int_0^1 x^pln^q(1/x)dxleint_0^1x^{p-q}dx= frac{1}{p-q+1}$$






real-analysis integration







share|cite|improve this question




















  • 4




    What about other cases? You clearly have not found all values.
    – xbh
    Nov 25 at 14:00






  • 1




    As $$int_{0}^{1}xlnleft(frac{1}{x}right)text{d}x=frac{1}{4}$$ your answer seems to be incomplete (it's only one of many examples which do not follow the condition you have found).
    – Galc127
    Nov 25 at 14:01












  • @xbh then what is the answer ?
    – jasmine
    Nov 25 at 14:06










  • @Galc127 okkss but u have take $p=q=1$
    – jasmine
    Nov 25 at 14:07






  • 1




    @jasmine, Oops, my bad... you can take $p=0,q=1$ and you have $$intlimits_{0}^{1}lnleft(frac{1}{x}right)text{d}x=1$$
    – Galc127
    Nov 25 at 14:11














1












1








1







find all value of $(p,q)$ for which integral



$$int_0^1 x^pln^q(1/x)dx$$



converges ?



My answer is $p > q-1$



Is It correct?



Proof : $$int_0^1 x^pln^q(1/x)dxleint_0^1x^{p-q}dx= frac{1}{p-q+1}$$






real-analysis integration







share|cite|improve this question















find all value of $(p,q)$ for which integral



$$int_0^1 x^pln^q(1/x)dx$$



converges ?



My answer is $p > q-1$



Is It correct?



Proof : $$int_0^1 x^pln^q(1/x)dxleint_0^1x^{p-q}dx= frac{1}{p-q+1}$$






real-analysis integration




real-analysis integration






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 26 at 15:33

























asked Nov 25 at 13:56









jasmine

1,523416




1,523416








  • 4




    What about other cases? You clearly have not found all values.
    – xbh
    Nov 25 at 14:00






  • 1




    As $$int_{0}^{1}xlnleft(frac{1}{x}right)text{d}x=frac{1}{4}$$ your answer seems to be incomplete (it's only one of many examples which do not follow the condition you have found).
    – Galc127
    Nov 25 at 14:01












  • @xbh then what is the answer ?
    – jasmine
    Nov 25 at 14:06










  • @Galc127 okkss but u have take $p=q=1$
    – jasmine
    Nov 25 at 14:07






  • 1




    @jasmine, Oops, my bad... you can take $p=0,q=1$ and you have $$intlimits_{0}^{1}lnleft(frac{1}{x}right)text{d}x=1$$
    – Galc127
    Nov 25 at 14:11














  • 4




    What about other cases? You clearly have not found all values.
    – xbh
    Nov 25 at 14:00






  • 1




    As $$int_{0}^{1}xlnleft(frac{1}{x}right)text{d}x=frac{1}{4}$$ your answer seems to be incomplete (it's only one of many examples which do not follow the condition you have found).
    – Galc127
    Nov 25 at 14:01












  • @xbh then what is the answer ?
    – jasmine
    Nov 25 at 14:06










  • @Galc127 okkss but u have take $p=q=1$
    – jasmine
    Nov 25 at 14:07






  • 1




    @jasmine, Oops, my bad... you can take $p=0,q=1$ and you have $$intlimits_{0}^{1}lnleft(frac{1}{x}right)text{d}x=1$$
    – Galc127
    Nov 25 at 14:11








4




4




What about other cases? You clearly have not found all values.
– xbh
Nov 25 at 14:00




What about other cases? You clearly have not found all values.
– xbh
Nov 25 at 14:00




1




1




As $$int_{0}^{1}xlnleft(frac{1}{x}right)text{d}x=frac{1}{4}$$ your answer seems to be incomplete (it's only one of many examples which do not follow the condition you have found).
– Galc127
Nov 25 at 14:01






As $$int_{0}^{1}xlnleft(frac{1}{x}right)text{d}x=frac{1}{4}$$ your answer seems to be incomplete (it's only one of many examples which do not follow the condition you have found).
– Galc127
Nov 25 at 14:01














@xbh then what is the answer ?
– jasmine
Nov 25 at 14:06




@xbh then what is the answer ?
– jasmine
Nov 25 at 14:06












@Galc127 okkss but u have take $p=q=1$
– jasmine
Nov 25 at 14:07




@Galc127 okkss but u have take $p=q=1$
– jasmine
Nov 25 at 14:07




1




1




@jasmine, Oops, my bad... you can take $p=0,q=1$ and you have $$intlimits_{0}^{1}lnleft(frac{1}{x}right)text{d}x=1$$
– Galc127
Nov 25 at 14:11




@jasmine, Oops, my bad... you can take $p=0,q=1$ and you have $$intlimits_{0}^{1}lnleft(frac{1}{x}right)text{d}x=1$$
– Galc127
Nov 25 at 14:11










1 Answer
1






active

oldest

votes


















2














Hint:
Let $lndfrac1x=u$ then
$$I=int_0^infty u^qe^{-(p+1)u} du$$
which may expressed as gamma function.






share|cite|improve this answer





















  • sir is there any other method?? as i didn't study gamma function .
    – jasmine
    Nov 25 at 14:06












  • @jasmine Then how about directly testing the convergence for Gamma function?
    – xbh
    Nov 25 at 14:09










  • @xbh then what is the correct answer , i need answer ????
    – jasmine
    Nov 25 at 14:16






  • 1




    @jasmine We have $lnalpha<alpha-1$ then $$I<int_0^1x^{p-q}(1-x)^{q} dx<int_0^1(1-x)^{q} dx=int_0^1x^q dx<infty$$ if $1+q>0$.
    – Nosrati
    Dec 2 at 17:36










  • thanks u sir @Nosrati
    – jasmine
    Dec 3 at 0:22











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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









2














Hint:
Let $lndfrac1x=u$ then
$$I=int_0^infty u^qe^{-(p+1)u} du$$
which may expressed as gamma function.






share|cite|improve this answer





















  • sir is there any other method?? as i didn't study gamma function .
    – jasmine
    Nov 25 at 14:06












  • @jasmine Then how about directly testing the convergence for Gamma function?
    – xbh
    Nov 25 at 14:09










  • @xbh then what is the correct answer , i need answer ????
    – jasmine
    Nov 25 at 14:16






  • 1




    @jasmine We have $lnalpha<alpha-1$ then $$I<int_0^1x^{p-q}(1-x)^{q} dx<int_0^1(1-x)^{q} dx=int_0^1x^q dx<infty$$ if $1+q>0$.
    – Nosrati
    Dec 2 at 17:36










  • thanks u sir @Nosrati
    – jasmine
    Dec 3 at 0:22
















2














Hint:
Let $lndfrac1x=u$ then
$$I=int_0^infty u^qe^{-(p+1)u} du$$
which may expressed as gamma function.






share|cite|improve this answer





















  • sir is there any other method?? as i didn't study gamma function .
    – jasmine
    Nov 25 at 14:06












  • @jasmine Then how about directly testing the convergence for Gamma function?
    – xbh
    Nov 25 at 14:09










  • @xbh then what is the correct answer , i need answer ????
    – jasmine
    Nov 25 at 14:16






  • 1




    @jasmine We have $lnalpha<alpha-1$ then $$I<int_0^1x^{p-q}(1-x)^{q} dx<int_0^1(1-x)^{q} dx=int_0^1x^q dx<infty$$ if $1+q>0$.
    – Nosrati
    Dec 2 at 17:36










  • thanks u sir @Nosrati
    – jasmine
    Dec 3 at 0:22














2












2








2






Hint:
Let $lndfrac1x=u$ then
$$I=int_0^infty u^qe^{-(p+1)u} du$$
which may expressed as gamma function.






share|cite|improve this answer












Hint:
Let $lndfrac1x=u$ then
$$I=int_0^infty u^qe^{-(p+1)u} du$$
which may expressed as gamma function.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Nov 25 at 14:01









Nosrati

26.4k62353




26.4k62353












  • sir is there any other method?? as i didn't study gamma function .
    – jasmine
    Nov 25 at 14:06












  • @jasmine Then how about directly testing the convergence for Gamma function?
    – xbh
    Nov 25 at 14:09










  • @xbh then what is the correct answer , i need answer ????
    – jasmine
    Nov 25 at 14:16






  • 1




    @jasmine We have $lnalpha<alpha-1$ then $$I<int_0^1x^{p-q}(1-x)^{q} dx<int_0^1(1-x)^{q} dx=int_0^1x^q dx<infty$$ if $1+q>0$.
    – Nosrati
    Dec 2 at 17:36










  • thanks u sir @Nosrati
    – jasmine
    Dec 3 at 0:22


















  • sir is there any other method?? as i didn't study gamma function .
    – jasmine
    Nov 25 at 14:06












  • @jasmine Then how about directly testing the convergence for Gamma function?
    – xbh
    Nov 25 at 14:09










  • @xbh then what is the correct answer , i need answer ????
    – jasmine
    Nov 25 at 14:16






  • 1




    @jasmine We have $lnalpha<alpha-1$ then $$I<int_0^1x^{p-q}(1-x)^{q} dx<int_0^1(1-x)^{q} dx=int_0^1x^q dx<infty$$ if $1+q>0$.
    – Nosrati
    Dec 2 at 17:36










  • thanks u sir @Nosrati
    – jasmine
    Dec 3 at 0:22
















sir is there any other method?? as i didn't study gamma function .
– jasmine
Nov 25 at 14:06






sir is there any other method?? as i didn't study gamma function .
– jasmine
Nov 25 at 14:06














@jasmine Then how about directly testing the convergence for Gamma function?
– xbh
Nov 25 at 14:09




@jasmine Then how about directly testing the convergence for Gamma function?
– xbh
Nov 25 at 14:09












@xbh then what is the correct answer , i need answer ????
– jasmine
Nov 25 at 14:16




@xbh then what is the correct answer , i need answer ????
– jasmine
Nov 25 at 14:16




1




1




@jasmine We have $lnalpha<alpha-1$ then $$I<int_0^1x^{p-q}(1-x)^{q} dx<int_0^1(1-x)^{q} dx=int_0^1x^q dx<infty$$ if $1+q>0$.
– Nosrati
Dec 2 at 17:36




@jasmine We have $lnalpha<alpha-1$ then $$I<int_0^1x^{p-q}(1-x)^{q} dx<int_0^1(1-x)^{q} dx=int_0^1x^q dx<infty$$ if $1+q>0$.
– Nosrati
Dec 2 at 17:36












thanks u sir @Nosrati
– jasmine
Dec 3 at 0:22




thanks u sir @Nosrati
– jasmine
Dec 3 at 0:22


















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