If d x (x+2) (x^2+1 ) =a |1+x^2 |+b ⁻¹ x+ 1 5 |x+2|+c , then

Step-by-step solution

Given, aligned & d x (x+2) (x^2+1 ) =a |1+x^2 |+b ⁻¹ x & + 1 5 |x+2|+c & Let 1 (x+2) (x^2+1 ) = A (x+2) + B x+C (x^2+1 ) & 1=A (x^2+1 )+(x+2)(B x+C) & 1=A x^2+A+B x^2+C x+2 B x+2 C & 1=(A+B) x^2+x(2 B+C)+A+2 C & aligned Here, A+B=0 , ; 2 B+C=0 and A+2 C=1 On solving them, we get A= 1 5 , B= -1 5 and C= 2 5 aligned & Therefore, d x (x+2) (x^2+1 ) & = ( 1 5(x+2) + - 5 5 x+ 2 5 x^2+1 ) d x & = 1 5 1 (x+2) d x- 1 5 x x^2+1 d x+ 2 5 d x 1+x^2 & = 1 5 |x+2|- 1 10 |1+x^2 |+ 2 5 ⁻¹ x+c & =- 1 10 (1+x^2 )+ 2 5 ⁻¹ x+ 1 5 |x+