Question: 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.

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 $$\int \frac{du}{u}$$
**Note:** Your comment at the end is incorrect. It is **not** true that
$$\int\frac{dx}{\arcsin x} = \ln(\arcsin x)+ C\tag{Wrong!}$$

This is, unfortunately, a common mistake. *Don't fall for it again!* 

While $$\int\frac{dx}{x} = \ln|x|+C$$ is true, in general, $$\int\frac{dx}{f(x)}$$ is **not** equal to $\ln|f(x)|+C$. If you differentiate $\ln|f(x)|$, you'll notice that you get $\frac{f'(x)}{f(x)}$. It is *precisely* because you have a $\frac{1}{\sqrt{1-x^2}}$ that you *do* get the natural logarithm of an arcsine.