Question

大神！求解答！！！

Answer 1

=∫[1 - 1/(1+x^2)] * arctanx * dx
=∫arctanx * dx - ∫arctanx * dx/(1+x^2)
=∫arctanx * dx - ∫arctanx * d(arctanx)
=∫θ*(secθ)^2 *dθ - 1/2 * (arctanx)^2 设 x = tanθ，则 dx = (secθ)^2 * dθ
=θ*tanθ - ∫ tanθ * dθ - 1/2 * (arctanx)^2
=θ*tanθ - ∫sinθ*dθ/cosθ - 1/2 * (arctanx)^2
=θ*tanθ + ∫d(cosθ)/cosθ - 1/2 * (arctanx)^2
=arctanx * x + ln(cosθ) - 1/2 * (arctanx)^2 + C
=x*arctanx - 1/2*ln(sec)^2 - 1/2 * (arctanx)^2 + C
=x*arctanx - 1/2*ln(1+x^2) - 1/2 * (arctanx)^2 + C