May I know how this integral was evaluated using hypergeometric function?

Question

I can not solve the following integral using the hypergeometric function:

$$\int_a^b (\sin x)^{(1/n)}dx$$

Wolframalpha showed the following result.

but I do not understand how Wolframalpha came to this result.

Thanks in advance.

Answer 1

The answer of WolframAlpha can be obtained by the change of variables $t=\cos^2 x, dt=-2\cos x\sin x\,dx$ and subsequent expansion of the initial integral: \begin{align} \int \sin^a x\,dx=&-\frac12\int\frac{\sin^{a-1}x}{\cos x}\left(-2\cos x\sin x\right)dx=\\=&-\frac12\int t^{-\frac12}\left(1-t\right)^{\frac{a-1}{2}}dt=\\ =&-\frac12\int t^{-\frac12}\left(\sum_{k=0}^{\infty}\frac{\left(\frac{1-a}{2}\right)_kt^k}{k!}\right)dt=\\ =&-\frac12\sum_{k=0}^{\infty}\frac{\left(\frac{1-a}{2}\right)_k t^{k+\frac12}}{k!\left(k+\frac12\right)}=-\sum_{k=0}^{\infty}\frac{\left(\frac{1-a}{2}\right)_k \left(\frac12\right)_kt^{k+\frac12}}{\left(\frac32\right)_kk!}=\\ =&-t^{\frac12}{}_2F_1\left(\frac12,\frac{1-a}{2};\frac32;t\right). \end{align} Here $(\alpha)_k=\frac{\Gamma(a+k)}{\Gamma(a)}$ denotes the Pochhammer's symbol.