Integrate xarctan(x^2)dx: Steps & Solution

[apiwowar]

the problem is find the integral of xarctan(x^2)dx

i set w = x^2, so 1/2dw = xdx

then i plug that into the integral to get

the integral of 1/2arctan(w)dw

so i let u = arctan(w) and dv = dw
so du = dw/(1+w^2) and v = w

so then the integral of udv = uv - integral of vdu

so 1/2(w*arctan(w) - integral of w * 1/(1+w^2)dw is what i end up with

but then if i would have to do integration by parts on the second integral

which gets me at

1/2(w*arctan(w) - wln(1+w^2) - integral of ln(1+w^2)dw

and that gets me stuck due to the having to take the antiderivative of the natural log

any help would be appreciated. and sorry if its hard to read

 

[vela]

You can do the second integral using the substitution u=w²+1. You don't need to integrate by parts.