I need a help integrating:
$\displaystyle
\int Arctan^2xdx
$
Any ideas?
Nah $\displaystyle \int Arctan x^2dx$ is easy :P. I need help with the other one and indeed it is an ugly one.
The best i got so far is this:
with partial intergration:
$\displaystyle u = (arctan x)^2, du = \frac{2(arctan x)}{(1 + x^2)}$
$\displaystyle dv = dx,v = x$
and im getting:
$\displaystyle \int arctan^2 x dx= x*(arctan^2 x) - \int\frac{2x*(arctan x)}{1 + x^2} dx$
Now im integrating $\displaystyle \int\frac{2x*(arctan x)}{1 + x^2}dx$ with partial:
$\displaystyle u = arctan x,du = \frac{1}{1 + x^2}$
$\displaystyle dv = \frac{2x}{1 + x^2},v = ln(1 + x^2)$
and im getting:
$\displaystyle \int\frac{2x*(arctan x)}{1 + x^2} dx
= ln(1 + x^2)*(arctan x) - \int\frac{ln(1 + x^2)}{1 + x^2} dx$
and im stuck at this last integral
I don't think so. I put your integral to check it with a software and its answer involves complex numbers.
There're lots of definite integrals which don't have elementary primitives, but workin' with integration limits one can find a suitable answer. In this case, I tried with some boundary limits but nothing worked out.