Question

arctanx/(x^2)的原函数

Answer 1

分部积分法。
∫arctanx/x²dx=∫arctanxd（-1/x）=-arctanx/x+∫darctanx/x=-arctanx/x+∫dx/（x（1+x²））=-arctanx/x+∫dx/x-∫xdx/（1+x²）=-arctanx/x+ln|x|-1/2ln（1+x²）+C

Answer 2

∫ arctanx / x² dx
= -∫ arctanx d(1/x)
= -(1/x)arctanx + ∫ (1/x)d(arctanx)
= -(1/x)arctanx + ∫ 1/[x(1+x²)] dx
= -(1/x)arctanx + ∫ (1/x)dx - ∫ x/(1+x²) dx
= -(1/x)arctanx + ln|x| - (1/2)ln(1+x²) + C

Note：
1/[x(1+x²)] = A/x + (Bx+C)/(1+x²)
1 = A(1+x²) + (Bx+C)x
1 = Ax²+A+Bx²+Cx
1 = (A+B)x²+Cx+A
1 = A
A+B = 0 => B=-1
C = 0
1/[x(1+x²)] = 1/x - x/(1+x²)