Question

第44题。。大一高数。。不定积分

Answer 1

(44)
x^3+1 = x(x^2+1) - x +1

∫(x^3+1)/(x^2+1)^2 dx
=∫ x/(x^2+1) dx - ∫(x-1)/(x^2+1)^2 dx
=∫ x/(x^2+1) dx - ∫x/(x^2+1)^2 dx + ∫dx/(x^2+1)^2
=(1/2)ln|x^2+1| + 1/[2(x^2+1)] + ∫dx/(x^2+1)^2

let
x=tany
dx=(secy)^2dy

∫dx/(x^2+1)^2
=∫(cosy)^2 dy
=(1/2)∫(1+cos2y) dy
=(1/2)[ y+(1/2)sin2y ] + C'
=(1/2)[ arctanx+ x/(x^2+1) ] + C'

∫(x^3+1)/(x^2+1)^2 dx
=(1/2)ln|x^^2+1| + 1/[2(x^2+1)] + ∫dx/(x^2+1)^2
=(1/2)ln|x^^2+1| + 1/[2(x^2+1)] + (1/2)[ arctanx+ x/(x^2+1) ] + C