Help on the following indefinite integral: $\int\big(\sqrt{1-t^2}\big)^{n-1}\mathrm{d}t$

Question

I would like to evaluate the following indefinite integral $$ \int\big(\sqrt{1-t^2}\big)^{n-1}\mathrm{d}t, $$ but -alas- I am not familiarized enough with this kind of integration. I have been suggested to use the substitution $t=\sin\varphi$, but I have some difficulties in applying that. Could you please help me? Thanks a lot!

Answer 1

By replacing $t$ with $\sin \phi$ we are left with: $$ I= \int \cos^n\phi\,d\phi \tag{1}$$ and this integral can be approached by considering the Fourier cosine series of the integrand function: $$\cos^n\phi = \frac{1}{2^n}\left(e^{i\phi}+e^{-i\phi}\right)^n = \frac{1}{2^n}\sum_{k=0}^{\lfloor n/2\rfloor}\binom{n}{k}2\cos((n-2k)\phi).\tag{2}$$ The RHS of $(2)$ is quite easy to integrate.