aligned & (1+x) (1+x^2 ) d x= (x+ x^2 2 + 1 2 ) & (1+x^2 )+g(x)+C, then g(x)= aligned

Step-by-step solution

Here (1+x) (1+x^2 ) d x= (1+x^2 ) 1 d x+ x (1+x^2 ) d x aligned = (1+x^2 ) x- ( 1 1+x^2 ) ( & (2 x) x d x & + x (1+x^2 ) d x aligned =x (1+x^2 )-2 ( x^2+1-1 1+x^2 ) d x+ x (1+x^2 ) d x aligned =x (1+x^2 )-2 [ (1- 1 1+x^2 ) d x ] & & + x (1+x^2 ) d x aligned aligned & =x (1+x^2 )-2 [ 1 d x- ( 1 1+x^2 ) d x ]+ x & =x (1+x^2 )-2 [x- ⁻¹ x ]+ x (1+x^2 ) d x & =x (1+x^2 )-2 x+2 ⁻¹ x x (1+x^2 ) d x aligned Now from x (1+ x ^2 ) dx let 1+ x ^2= t 2 x d x=d t aligned & = 1 2 t d t = 1 2 [ t t - ( t t t ) dt ] & =-[ t lot t