Intgeration by parts problem

Nov 2018
24
1
Iran
Hello, I can't solve this integral which is in Pikunov's book . It says use integration by parts, but I can't solve it.
Problem is $$\int \frac{x \arctan(x)}{(1+x^2)^2} \,dx$$ thank U by the way.
 
Dec 2015
972
128
Earth
Start with \(\displaystyle x=\tan(v)\).
 
Dec 2015
972
128
Earth
\(\displaystyle dx=sec^2 (v) dv \; \)
$ \int \frac{x \arctan(x)}{(1+x^2)^2} \,dx =\int v\tan(v)\cos^4 (v) sec^2 (v) dv =\int v\sin(v)\cos(v)dv=
\dfrac{\sin\left(2v\right)-2v\cos\left(2v\right)}{8}+C

$.
$ v\sin(v)\cos(v)dv=\dfrac{2v}{4}\sin(2v) dv $.
 
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