Integrate $\sin^n{x}$

Question

How do you integrate:

$$\int(\sin^n{x}) dx$$

The link to WolframAlpha : (Integration Answer)

No definite limits...

What is that hypergeometric function in that answer. Please help!

Thanks

Answer 1

Using binomial series (link), we have: $$(1-\cos^2x)^{\frac{n-1}{2}} = 1 - \frac{n-1}{2}\cos^2x + \frac{n-1}{2}\frac{n-3}{4}\cos^4x-\ldots. $$

Then:

$$ \int\sin^nx dx = \int\sin x (1-\cos^2x)^{\frac{n-1}{2}} dx $$ $$ = \int\sin x \left(1 - \frac{n-1}{2}\cos^2x + \frac{n-1}{2}\frac{n-3}{4}\cos^4x-\ldots\right) dx $$ $$ = \int \left(\sin x + \frac{1-n}{2}\cos^2x \sin x+ \frac{1-n}{2}\frac{3-n}{4}\cos^4x\sin x -\ldots\right) dx $$ $$ = -\cos x - \frac{1-n}{2}\cos^3x/3 - \frac{1-n}{2}\frac{3-n}{4}\cos^5x/5 -\ldots$$ $$ = -\cos x \cdot({\rm hypergeometric \; series}).$$