Improper integral with parameter. For what value of a does the integral converge?

Question

i just cant figure it out.... Please explain...

$ \int\limits_{\scriptsize 0}^{\scriptsize \pi}{\cos\left(x\right)\,\sin^{a}\left(x\right)}{\;\mathrm{d}x} $

Answer 1

Let $u = \sin(x)$ then $du=\cos(x) dx$.

$$\int_c^d \cos(x) \sin^a(x) dx = \int_c^d u^a du=\frac{u^{a+1}}{a+1}\bigg|_c^d= \frac{(\sin(x))^{a+1}}{a+1}\Bigg|_c^d.$$

So the integral doe not converge when $a=-1$. So the integral converges for all $a>-1$. For $a<-1$, the integral is not defined (blow up) on the boundaries, namely $x=0$, and $x=\pi$. So the integral converges for $a \in (-1,\infty)$