Integrating $x^c (1-x)^d$ from 0 to 1 using the gamma function












-1












$begingroup$


I am trying to solve the following integral:
$$
int_0^1 x^c (1-x)^d dx
$$
for some $c, d in mathbb{R}$
I know I have to use the gamma function, I have tried using the substitutions $u = lnfrac{1}{x}$ as well as $u = lnfrac{1}{1-x}$ to try to make this appear as a gamma function but neither have gotten me closer to an answer.






integration gamma-function







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  • 4




    $begingroup$
    en.wikipedia.org/wiki/Beta_function
    $endgroup$
    – Lord Shark the Unknown
    Jan 2 at 11:17






  • 1




    $begingroup$
    Try $x=sin^2(t)$
    $endgroup$
    – Claude Leibovici
    Jan 2 at 11:19
















-1












$begingroup$


I am trying to solve the following integral:
$$
int_0^1 x^c (1-x)^d dx
$$
for some $c, d in mathbb{R}$
I know I have to use the gamma function, I have tried using the substitutions $u = lnfrac{1}{x}$ as well as $u = lnfrac{1}{1-x}$ to try to make this appear as a gamma function but neither have gotten me closer to an answer.






integration gamma-function







share|cite|improve this question









$endgroup$








  • 4




    $begingroup$
    en.wikipedia.org/wiki/Beta_function
    $endgroup$
    – Lord Shark the Unknown
    Jan 2 at 11:17






  • 1




    $begingroup$
    Try $x=sin^2(t)$
    $endgroup$
    – Claude Leibovici
    Jan 2 at 11:19














-1












-1








-1





$begingroup$


I am trying to solve the following integral:
$$
int_0^1 x^c (1-x)^d dx
$$
for some $c, d in mathbb{R}$
I know I have to use the gamma function, I have tried using the substitutions $u = lnfrac{1}{x}$ as well as $u = lnfrac{1}{1-x}$ to try to make this appear as a gamma function but neither have gotten me closer to an answer.






integration gamma-function







share|cite|improve this question









$endgroup$




I am trying to solve the following integral:
$$
int_0^1 x^c (1-x)^d dx
$$
for some $c, d in mathbb{R}$
I know I have to use the gamma function, I have tried using the substitutions $u = lnfrac{1}{x}$ as well as $u = lnfrac{1}{1-x}$ to try to make this appear as a gamma function but neither have gotten me closer to an answer.






integration gamma-function




integration gamma-function






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share|cite|improve this question




share|cite|improve this question










asked Jan 2 at 11:11









Abdelrahman SamehAbdelrahman Sameh

32




32








  • 4




    $begingroup$
    en.wikipedia.org/wiki/Beta_function
    $endgroup$
    – Lord Shark the Unknown
    Jan 2 at 11:17






  • 1




    $begingroup$
    Try $x=sin^2(t)$
    $endgroup$
    – Claude Leibovici
    Jan 2 at 11:19














  • 4




    $begingroup$
    en.wikipedia.org/wiki/Beta_function
    $endgroup$
    – Lord Shark the Unknown
    Jan 2 at 11:17






  • 1




    $begingroup$
    Try $x=sin^2(t)$
    $endgroup$
    – Claude Leibovici
    Jan 2 at 11:19








4




4




$begingroup$
en.wikipedia.org/wiki/Beta_function
$endgroup$
– Lord Shark the Unknown
Jan 2 at 11:17




$begingroup$
en.wikipedia.org/wiki/Beta_function
$endgroup$
– Lord Shark the Unknown
Jan 2 at 11:17




1




1




$begingroup$
Try $x=sin^2(t)$
$endgroup$
– Claude Leibovici
Jan 2 at 11:19




$begingroup$
Try $x=sin^2(t)$
$endgroup$
– Claude Leibovici
Jan 2 at 11:19










1 Answer
1






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oldest

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2












$begingroup$

Hint:



$$int_0^1 t^{x-1}(1-t)^{y-1}dt=B(x,y)=frac{Gamma(x)Gamma(y)}{Gamma(x+y)}$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    But is there a way to do it using only the gamma function and its definition?
    $endgroup$
    – Abdelrahman Sameh
    Jan 2 at 12:12










  • $begingroup$
    @AbdelrahmanSameh A proof of the Beta Function comes from using the definition of the gamma function. I believe it's on Wikipedia
    $endgroup$
    – Frank W.
    Jan 2 at 19:03











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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









2












$begingroup$

Hint:



$$int_0^1 t^{x-1}(1-t)^{y-1}dt=B(x,y)=frac{Gamma(x)Gamma(y)}{Gamma(x+y)}$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    But is there a way to do it using only the gamma function and its definition?
    $endgroup$
    – Abdelrahman Sameh
    Jan 2 at 12:12










  • $begingroup$
    @AbdelrahmanSameh A proof of the Beta Function comes from using the definition of the gamma function. I believe it's on Wikipedia
    $endgroup$
    – Frank W.
    Jan 2 at 19:03
















2












$begingroup$

Hint:



$$int_0^1 t^{x-1}(1-t)^{y-1}dt=B(x,y)=frac{Gamma(x)Gamma(y)}{Gamma(x+y)}$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    But is there a way to do it using only the gamma function and its definition?
    $endgroup$
    – Abdelrahman Sameh
    Jan 2 at 12:12










  • $begingroup$
    @AbdelrahmanSameh A proof of the Beta Function comes from using the definition of the gamma function. I believe it's on Wikipedia
    $endgroup$
    – Frank W.
    Jan 2 at 19:03














2












2








2





$begingroup$

Hint:



$$int_0^1 t^{x-1}(1-t)^{y-1}dt=B(x,y)=frac{Gamma(x)Gamma(y)}{Gamma(x+y)}$$






share|cite|improve this answer









$endgroup$



Hint:



$$int_0^1 t^{x-1}(1-t)^{y-1}dt=B(x,y)=frac{Gamma(x)Gamma(y)}{Gamma(x+y)}$$







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Jan 2 at 11:51









DarkraiDarkrai

6,2971441




6,2971441












  • $begingroup$
    But is there a way to do it using only the gamma function and its definition?
    $endgroup$
    – Abdelrahman Sameh
    Jan 2 at 12:12










  • $begingroup$
    @AbdelrahmanSameh A proof of the Beta Function comes from using the definition of the gamma function. I believe it's on Wikipedia
    $endgroup$
    – Frank W.
    Jan 2 at 19:03


















  • $begingroup$
    But is there a way to do it using only the gamma function and its definition?
    $endgroup$
    – Abdelrahman Sameh
    Jan 2 at 12:12










  • $begingroup$
    @AbdelrahmanSameh A proof of the Beta Function comes from using the definition of the gamma function. I believe it's on Wikipedia
    $endgroup$
    – Frank W.
    Jan 2 at 19:03
















$begingroup$
But is there a way to do it using only the gamma function and its definition?
$endgroup$
– Abdelrahman Sameh
Jan 2 at 12:12




$begingroup$
But is there a way to do it using only the gamma function and its definition?
$endgroup$
– Abdelrahman Sameh
Jan 2 at 12:12












$begingroup$
@AbdelrahmanSameh A proof of the Beta Function comes from using the definition of the gamma function. I believe it's on Wikipedia
$endgroup$
– Frank W.
Jan 2 at 19:03




$begingroup$
@AbdelrahmanSameh A proof of the Beta Function comes from using the definition of the gamma function. I believe it's on Wikipedia
$endgroup$
– Frank W.
Jan 2 at 19:03


















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