How to calculate $\int_{0}^{\pi/4}\sin^{n+1}\theta\cos^{n-1}\theta d\theta$?

Question

My approach is to use the properties of the beta function. But the limits of integration are from zero to $\frac{\pi}{4}$. We know that

$$ B(m,n) = 2\int_{0}^{\pi/2}\sin^{2m-1}\theta \cos^{2n-1}\theta d\theta = \dfrac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)} $$

Thanks for any help.

Answer 1

Hint: Use substitution $\theta=\dfrac{u}{2}$ then continue with $$\dfrac{1}{2^{n+1}}\int_0^{\pi/2}\sin^{n-1}u(1-\cos u)du=\dfrac{\beta(\frac{n}{2},\frac12)-\beta(\frac{n}{2},1)}{2^{n+2}}$$