设x=tant. t∈[0, π/4].
则 ∫ ln(1+x)/(1+x^2) dx.
=∫ ln(1+tant)/ (1+tant ^2) *sect^2 dt.
=∫ ln(1+tant) dt.
=∫ ln(sint+tant)-ln(cost) dt.
=∫ ln(√2 *(sin(t+π/4)))-ln(cost) dt.
=∫ 1/2 *ln2+ln(sin(t+π/4))-ln(cost) dt.(t从0->π/4).
=π*ln2/8+∫ ln(sint) dt (t从π/2->π/4) -∫ ln(sint) dt (t从π/2->π/4).
=π*ln2/8.