Evaluate this indefinite inegral $\int \sin^{e}x \,dx$

Question

How to evaluate this indefinite integral

$$\int \sin^{e}x\, dx$$

I evaluate from wolfram aplha but i didn't get it I have no idea from where I should start. Please give me hint.

Answer 1

The graph of this function can be computed from this https://www.symbolab.com/graphing-calculator My idea is first to find the area of each non-zero part in it and then take sum and see its convergence

Answer 2

The following is not a solution, it is just an idea. I'm not sure if it's the right path: $$I = \int \sin^{e}x\, dx$$

Apply the substitution $$u=\sin(x) \Leftrightarrow x = \arcsin(u)$$$$ du = \cos(x)dx = \cos(\arcsin(u))\,dx = \sqrt{1 - u^2}\,dx \implies dx = \frac{1}{\sqrt{1 - u^2}}\, du$$ Also

$$I = \int \frac{u^e}{\sqrt{1 - u^2}} \, du $$
Multiply by the conjugate und we get $$I = \int\frac{u^e\sqrt{1-u^2}}{1-u^2}\,du$$