<s>[INST] Integration by parts - hint I'm stuck on a passage on my textbook:
$$ \int \frac{1}{(1+t^2)^3} dt = \frac{t}{4(t^2+1)^2}+\frac{3}{4} \int \frac{1}{(t^2+1)^2} dt$$
I know that it should be easy but I just can't figure out what is the product of functions considered in this integration by parts..
can you help me?
thanks a lot
 [/INST] $$\int \dfrac {1}{(t^2+1)^3}\,dt = \int \dfrac {(1+t^2) - t^2}{(t^2+1)^3}\, dt = \int \dfrac {1}{(t^2+1)^2}\,dt + \dfrac{1}{4}\int t\, d(t^2+1)^{-2}$$$$ = \int \dfrac {1}{(t^2+1)^2}\,dt + \dfrac{1}{4}\dfrac {t}{(t^2+1)^2} - \dfrac{1}{4}\int  \dfrac{1}{(t^2+1)^2}\,dt $$$$ = \dfrac{1}{4}\dfrac {t}{(t^2+1)^2} + \dfrac{3}{4}\int  \dfrac{1}{(t^2+1)^2}\,dt  $$
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