<s>[INST] Find indefinite integral $\int \frac{\arcsin (x)}{x^2}\, \mathrm{d}x$ $$\int \frac{\arcsin (x)}{x^2}\, \mathrm{d}x$$
$$\frac {1}{x^2\sqrt{1-x^2}}+2\int \frac{1}{x^3\sqrt{1-x^2}}\, \mathrm{d}x$$
I try to integrate by parts method, but its doesnt want to be solved. I try to substitue $x=\sin u \mathrm{d}x=\cos u$ but failed somewhere
 [/INST] Consider 
$$
\int\frac{\arcsin(x)}{x^2}dx.
$$
We may use integration by parts where
\begin{align*}
u&=\arcsin(x)&dv&=\frac{dx}{x^2}\\
du&=\frac{dx}{\sqrt{1-x^2}}&v&=-\frac{1}{x}.
\end{align*}
In this case,
$$
\int\frac{\arcsin(x)}{x^2}dx=-\frac{\arcsin(x)}{x}-\int-\frac{dx}{x\sqrt{1-x^2}}=-\frac{\arcsin(x)}{x}+\int\frac{dx}{x\sqrt{1-x^2}}.
$$
Focusing on the last term, we would like to use $u$-substitution, $u^2=1-x^2$ so that $2udu=-2xdx$.  In other words, $dx=-\frac{udu}{x}$.  Note that in this substitution, it only makes sense when $1-x^2\geq 0$, i.e., $|x|\leq 1$ and we may assume that $u$ is positive.  Using this substitution, we get the integral
$$
\int\frac{1}{x\sqrt{u^2}}\cdot\frac{-udu}{x}=-\int\frac{u}{|u|}\frac{du}{x^2}=-\int\frac{du}{1-u^2}=-\int\frac{du}{1-u^2}.
$$
We can now use partial fractions decomposition noting that
$$
\frac{1}{1-u^2}=\frac{1}{2(1-u)}+\frac{1}{2(1+u)}
$$
This ends up being
$$
-\left(-\frac{1}{2}\ln|1-u|+\frac{1}{2}\ln|1+u|\right).
$$
Now, finally, combine everything and you have your integral (near $x=0$).  After substitution, one gets 
$$
-\frac{\arcsin(x)}{x}+\frac{1}{2}\ln\left|1-\sqrt{1-x^2}\right|-\frac{1}{2}\ln\left|1+\sqrt{1-x^2}\right|
$$
whose derivative is $\frac{\arcsin(x)}{x^2}$ (checked with Maple).
Comparing this with the accepted answer, we can use log rules to get
$$
-\frac{\arcsin(x)}{x}+\frac{1}{2}\ln\left|1-\sqrt{1-x^2}\right|-\frac{1}{2}\ln\left|1+\sqrt{1-x^2}\right|=-\frac{\arcsin(x)}{x}+\frac{1}{2}\ln\left|\frac{1-\sqrt{1-x^2}}{1+\sqrt{1-x^2}}\right|.
$$
Multiplying the numerator and denominator of the argument to the logarithm by the conjugate of the numerator, we get
$$
-\frac{\arcsin(x)}{x}+\frac{1}{2}\ln\left|\frac{1-(1-x^2)}{(1+\sqrt{1-x^2})^2}\right|=-\frac{\arcsin(x)}{x}+\frac{1}{2}\ln\left|\frac{x}{1+\sqrt{1-x^2}}\right|^2.
$$
Using log rules the $2$'s cancel and we can break up the logarithm, leaving 
$$
-\frac{\arcsin(x)}{x}+\ln\left|\frac{x}{1+\sqrt{1-x^2}}\right|=-\frac{\arcsin(x)}{x}-\ln\left|1+\sqrt{1-x^2}\right|+\ln\left|x\right|
$$
Finally, remember $+C$.
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