Integrate x^2/(1+x^2)dx

Unlike if we had just x in the numerature, here the chain rule will do us no good.  Integration by parts in this case also is little help - we end up with expressions at least as difficult as we started with.  For example:

Int x²/(1+x²)dx = x²tan^-1x - Int 2x tan^-1x dx

In this case, a very simple approach is to simply add and subtract 1 from the denominator:

Int x²/(1+x²)dx = Int (x²+1-1)/(1+x²)dx= Int (1+x²)/(1+x²) - 1/(1+x²)dx = Int 1 - 1/(1+x²)dx

From here, do you know know you to solve the problem?