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?
LH
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