Question

求(1+3x^2)/x^2*(x^2+1)的不定积分

Answer 1

∫ { (1+3x^2)/[x^2*(x^2+1) } dx
=∫ {3/(x^2+1) + 1/[x^2.(x^2+1)] } dx
=3arctanx + ∫dx/[x^2.(x^2+1)]
=3arctanx + ∫ [1/x^2 -1/(x^2-1) ]dx
=3arctanx -1/x - ∫ 1/(x^2-1) dx
let
x= secy
dx= secy tany dy
∫ 1/(x^2-1) dx
=∫ (secy/tany) dy
=∫ cscy dy
=-ln|cscy -coty | + C'
=-ln|x/ √(x^2-1) - 1/√(x^2-1) | + C'
∫ { (1+3x^2)/[x^2*(x^2+1) } dx
=3arctanx -1/x - ∫ 1/(x^2-1) dx
=3arctanx -1/x +ln|x/ √(x^2-1) - 1/√(x^2-1) | + C