Question

这个积分怎么算啊

Answer 1

let 2x^3/[(2x^2+1)(x^2+1)]≡(Ax+B)/(2x^2+1)+ (Cx+D)/(x^2+1) => 2x^3 ≡ (Ax+B)(x^2+1)+ (Cx+D)(2x^2+1) coef. of x^3 : A+2C =2 (1) coef. of x^2 : B+2D =0 (2) coef. of x : A+C=0 (3) coef. of constnat : B+D =0 (4) (1)-(3) C=2 from (1) A+2C =2 A+4 =2 A=-2 (2)-(4) D=0 from (2) B=0 2x^3/[(2x^2+1)(x^2+1)]≡ -2x/(2x^2+1)+ 2x/(x^2+1) ∫ 2x^3/[(2x^2+1)(x^2+1)] dx =∫[-2x/(2x^2+1)+ 2x/(x^2+1)] dx = -(1/2)ln|2x^2+1| + ln|x^2+1| + C