求定积分 (0,1)∫ln(1+x^2)dx
2个回答

用分部积分法:

∫(0,1)ln(1+x^2)dx

=x ln(1+x^2)|(0,1)- ∫(0,1)xdln(1+x^2)

=ln2 - ∫(0,1)2x^2/(1+x^2)dx

=ln2 - 2∫(0,1)[1 - 1/(1+x^2)]dx

=ln2 - 2(x - arctanx))|(0,1)

=ln2 - 2 + π/2