Integrate $\int x^2\sin^{-1}\left ( \frac{\sqrt{a^2-x^2} }{b} \right ) dx$ 

I have  already done some progress for this integral at mathexchange

The objective is to compute the following definite integral:

$\int^w_0 x^2\sin^{-1}\left ( \frac{\sqrt{a^2-x^2} }{b} \right ) dx$

Once the indefinite one is done you can easily proceed with the limits. The constants $a$ and $b$ are non-zero positive, and $x<a$.

  • I do not want a complex form such as the one from Wolfram Alpha. I need the working out of it too.

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  • Please leave a comment if you need any clarifications.

  • Thank you Philip for the response. Sorry in advance if I sound ignorant (I am haven't got much knowledge on these elliptic integrals). Can the result $I(x)$ from the Wolfram Alpha page be plotted in a graph? (provided a and b).

  • Yes, you may graph them by Matlab: https://www.mathworks.com/help/matlab/ref/ellipke.html

  • All functions appearing in the formula are basic, all you need is to be able to graph the elliptic integrals of first and second kind.

The answer is accepted.