I'm supposed do find some integrals. I am stuck with two of them.
The first one: $\displaystyle\int \operatorname{arcsin}\left(\frac{1}{x}\right) dx$
I have already integrated by parts having:
$$\displaystyle x \operatorname{arcsin}\left(\frac{1}{x}\right) + \int \frac{1}{\sqrt{1-x^2} \cdot x} dx$$
I tried further integration with substitution $\sqrt{1-x^2} = t$, but i wasn't able to get any result of it.
It would be nice if anyone could help my with the further integral.
Best regards!
Best Answer
As both user49685 and Claude Leibovici have pointed out, check the derivative of $\arcsin(\frac{1}{x})$.
Once you have fixed the error, for the resulting integral try the change of variable $x=\sec\theta$, noting that $\tan^2\theta=\sec^2\theta-1$.