I am having a problem solving an integral. I am stuck in an infinite loop. Integral is:
$$\int{\frac{dx}{\sqrt{1-x^2}\arcsin{x}}}$$
I have separated it in dv and u on this way:
$$u = \frac{dx}{\sqrt{1-x^2}}$$
$$dv = \frac{1}{\arcsin{x}}$$
And the using:
$$u v – \int{v \, du}$$
I get again:
$$\int{\frac{dx}{\sqrt{1-x^2}\arcsin{x}}}$$
I dont know, but probably, I am doing something wrong. I am new at solving Integrals so I am learning 🙂 According to my book the result should be:
$$\ln({\arcsin{x}})-C$$
And it will be true if I didn't had $$\sqrt{1-x^2}$$ but on this way I have no idea.
Best Answer
Do not use Integration by Parts. Use $u$-substitution. Let $u=\sin^{-1}(x)$. Then $du=dx/\sqrt{1-x^2}$
So now your integral is $$\int \frac{du}{u}$$