Beta function proof

**simplependulum** · June 14th 2009, 12:34 AM

Prove that $\int_{0}^{\pi} sin^n(x) ~dx = B( \frac{n+1}{2} , \frac{1}{2} )$
where n is a real number ( guys you can prove whether n can be complex ... )

Then use this result to solve a Physics problem :
Caculate the time taken for a particle to travel from the top of a unit semicircle to the bottom . ( under gravitational force )

**Amer** · June 14th 2009, 01:32 AM

Originally Posted by simplependulum

Prove that $\int_{0}^{\pi} sin^n(x) ~dx = B( \frac{n+1}{2} , \frac{1}{2} )$
where n is a real number ( guys you can prove whether n can be complex ... )

Then use this result to solve a Physics problem :
Caculate the time taken for a particle to travel from the top of a unit semicircle to the bottom . ( under gravitational force )

you know that

$\beta (p,q) = 2\int_{0}^{\frac{\pi}{2}} sin^{2p-1}x cos^{2q-1}x dx$

you have

$\int_{0}^{\pi} sin^nx dx = \int_{0}^{\frac{\pi}{2}} sin^nx dx + \int_{\frac{\pi}{2}}^{\pi} sin^nx dx$

the first one equal

$\left(\frac{{\color{red}2}}{2}\right) {\color{red}\int_{0}^{\frac{\pi}{2}} sin^n x dx}=\frac{\beta(\frac{n+1}{2} , \frac{1}{2})}{2}$

since the red term equal beta and you have $2p-1 = n \Rightarrow p=\frac{n+1}{2}$ and $2q-1 = 0 \Rightarrow q=\frac{1}{2}$

the second integral

$\int_{\frac{\pi}{2}}^{\pi} sin^nx dx$ let $x=y+\frac{\pi}{2} \Rightarrow dx=dy$

$\int_{0}^{\frac{\pi}{2}} sin^n(y+\frac{\pi}{2}) dy =\int_{0}^{\frac{\pi}{2}} \left[siny(cos(\frac{\pi}{2}))+cosy(sin(\frac{\pi}{2}))\ right]^{n} dy$

$\Rightarrow \left(\frac{2}{2}\right)\int_{0}^{\frac{\pi}{2}} (cosy)^{n} dy =\frac{\beta(\frac{n+1}{2} , \frac{1}{2} )}{2}$

find the sum of the first and the second integarl you will have the answer

**Moo** · June 14th 2009, 01:34 AM

Hello,

Consider this formula of the beta function :

$\beta(x,y)=2\int_0^{\pi/2} \sin^{2x-1}\theta\cos^{2y-1}\theta ~d\theta$ (which you can get by a substitution $t=\sin^2\theta$ in the common formula of the beta function)

Then $\beta\left(\frac{n+1}{2},\frac 12\right)=2\int_0^{\pi/2} \sin^n\theta ~d\theta$

And since $\sin\left(\tfrac\pi 2-x\right)=\sin\left(\tfrac\pi 2+x\right) ~\dots$

**mr fantastic** · June 14th 2009, 04:03 AM

Originally Posted by simplependulum

Prove that $\int_{0}^{\pi} sin^n(x) ~dx = B( \frac{n+1}{2} , \frac{1}{2} )$
where n is a real number ( guys you can prove whether n can be complex ... )

Then use this result to solve a Physics problem :
Caculate the time taken for a particle to travel from the top of a unit semicircle to the bottom . ( under gravitational force )

Assuming the particle slides frictionlessly under gravity down the semicircle, the particle will leave the circle after the motion has gone through a polar angle equal to $\theta = \cos^{-1} \frac{2}{3}$ . So I'm not sure how the particle can end up at the bottom of the semicircle ....

Math Help - Beta function proof

LinkBack

Thread Tools

Display

Beta function proof

Similar Math Help Forum Discussions

beta proof

Beta function?

Beta Distribution Proof

beta function

beta distribution probability density function proof

Search Tags