₀^ 2 _ n=0 ^4 ( n 4 +x ) x+ x d x=

Step-by-step solution

Let f= ₀^ 2 _ n=0 ^4 4 +x x+ x d r since _ x=1 ^4 ( ex 4 +x )=(0+1+2+3+4) 4 +(1+1+1+1+1) x= 5 2 +5 x=5 ( 2 +x ) Hence, f= ₀^ 2 5 ( 2 +x ) d x 2 ( 1 2 x+ 1 2 x ) I= 5 2 ₀^ 2 ( 2 +x ) (x- 4 ) d x ...(i) Replacing (x) by ( 2 -x ) we get I= 5 2 ₀^ 2 [ 2 + ( 2 -x ) ] [ ( 2 -x )- 4 ] d x I= 5 2 ₀^ 2 [ -x] (x- 4 ) d x ...(ii) (- )= Adding equations (i) & (ii), we get 2 I= 5 2 ₀^ 2 ( 3 2 ) (x- 4 ) d x aligned & I= 15 4 2 [ | (x- 4 )+ (x- 4 ) | ]₀^ 2 & I= 15 4 2 | 2 +1 2 -1 |= 15 4 2 |( 2 +1)^2 | & I= 15 2 2 | 2 +1| aligned