Definite integral of a power of the sine or cosine

Question

Observing few results I have made one conclusion. Please tell me if it is correct or not. When limits of integration are $0$ to $n \pi$ and if function is $\sin^m x$ or $\cos^m x$, then answer is $0$ if $m$ is odd and if $m$ is even we can simplify it as $2n$ times the integration of given function with limits $0$ and $\frac{\pi}{2}$ so that I can use Walli's formula.

Answer 1

I think the integration interval you're looking for may be from $0$ to $2k\pi.$

If $m$ is odd then \begin{align} \int_0^{2k\pi} \sin^m t\,dt &= k \int_0^{2\pi} \sin^m t\,dt = 0, \\ \int_0^{2k\pi} \cos^m t\,dt &= k \int_0^{2\pi} \cos^m t\,dt = 0 \end{align} as special cases of your earlier question, Definite integral of the product of powers of the sine and cosine; that is, we can apply the identity $$ \int_0^{2\pi} \sin^m t \cos^n t\,dt = 0, $$ either with $m$ odd and $n= 0$ or with $n$ odd and $m = 0.$

For even $m$ you are correct that you can use the Wallis Cosine Formula for the interval $0$ to $\frac\pi2;$ the integral from $0$ to $2k\pi$ (or even the integral from $0$ to $\frac{k\pi}{2}$) is just an integer number of "copies" of the integral from $0$ to $\frac\pi2.$