₀^ / 2 x x 1+ x d x is equal to

Step-by-step solution

aligned Let I & = ₀^ / 2 x x 1+ x d x & = ₀^ / 2 x(1+ x-1) 1+ x d x I & = ₀^ / 2 ( x(1+ x) 1+ x - x 1+ x ) d x aligned aligned & = ₀^ / 2 x d x- ₀^ / 2 x 1+ x d x=[ x]₀^ / 2 -I₁ I₁ & = ₀^ / 2 x 1+ x d x aligned Let 1+ x=t aligned x d x & =d t I₁ & = ₀^ / 2 d t t =[ t]₀^ / 2 =[ (1+ x)]₀^ / 2 & = (1+1)- (1+0)= 2 aligned Now, from Eq. (i), we have I=[1-0]- 2=1- 2