7.设f arctanx x2，求f x .
解 由f arctanx x2知f x tan2x，因此
f x tan2xdx sec2x 1 dx tanx x C.
另解：因为 f arctanx f arctanx
1
1 x2
，所以 f arctanx 1 x2 f arctanx
. 因f arctanx x2，故
1 x2
f arctanx x, f arctanx
x221 x2.
于是
f arctanx x2 1 1 x2dx 1 1 x2 dx x arctanx C tanarctanx arctanx C,f u tanu u C,f x tanx x C.