∫arctanx/(x^2(1+x^2))dx

∫arctanx/(x^2(1+x^2))dx
∫arctanx/(x^2(1+x^2))dx

∫arctanx/(x^2(1+x^2))dx
令x = tanz,dx = sec²z dz
∫ arctanx/[x²(1 + x²)] dx
= ∫ z/(tan²zsec²z) * (sec²z dz)
= ∫ zcot²z dz
= ∫ z(csc²z - 1) dz
= ∫ zcsc²z dz - ∫ z dz
= ∫ z d(- cotz) - ∫ z dz
= - zcotz + ∫ cotz dz - ∫ z dz
= - zcotz + ln|sinx| - z²/2 + C
= - arctan(x)/x + ln|x/√(1 + x²)| - (1/2)(arctanx)² + C