don't know hot to integrate and do this question. I integrated it by parts then got to integrate (x^2/ 2 + 2x^2) again..
Not sure what you mean, but parts is right (for question a). Integrate x, differentiate arctan x, then integrate 1/2 x^2 all over 1 + x^2, which is equal to
$\displaystyle \frac{1}{2}\ \frac{1 + x^2 - 1}{1 + x^2}$
which you can split into
$\displaystyle \frac{1}{2}\ -\ \frac{1}{2}\ \frac{1}{1 + x^2}$
Just in case a picture helps...
More in a tic
By re-writing the numerator in order to split it as shown above... and now below...
Spoiler:
... is the product rule. Straight lines differentiate downwards (integrate up) with respect to x.
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Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!
You know, the top bit.
$\displaystyle \frac{1}{2}\ \frac{x^2}{1 + x^2}$
$\displaystyle =\ \frac{1}{2}\ \frac{1 + x^2 - 1}{1 + x^2}$
$\displaystyle =\ \frac{1}{2}\ \frac{1 + x^2}{1 + x^2}\ -\ \frac{1}{2}\ \frac{1}{1 + x^2}$
$\displaystyle =\ \frac{1}{2}\ -\ \frac{1}{2}\ \frac{1}{1 + x^2}$
It's a trick to by-pass long division, which you should nonetheless know.