₀^ 2 x (x) 1+ (x) d x=

Step-by-step solution

I= ₀^ 2 x x 1+ x d x On applying property ₀^a f(x) d x= ₀^a f(a-x) d x , we get I= ₀^ 2 (2 -x) x 1+ x d x (ii) (2 -x)= x On adding Eqs. (i) and (ii), we get aligned 2 I & =2 ₀^ 2 x 1+ x d x I & =2 ₀^ x 1+ x d x=2 ₀^ (1- 1 1+ x ) d x & =2 ₀^ (1- ^2 x 2 2 ) d x=2 [x- x 2 ]₀^ aligned