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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!

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    $\begingroup$ I think you should recheck your work on integrating by parts, it doesn't look correct to me. Please note that $(\arcsin x)' = \frac{1}{\sqrt{1-x^2}}$. $\endgroup$
    – user49685
    Mar 9 '14 at 18:55
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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 $\tan^2\theta=\sec^2\theta-1$.

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Hint

As user49685 pointed out, the derivative of $\arcsin(\frac{1}{x})$ is not what you wrote. Fix it and consider a change of variable such as $x=\cosh(y)$.

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