₀^ 2 x d x 1+ x x =

Step-by-step solution

₀^ 2 x d x 1+ x x I= ₀^ 2 ( 2 -x ) 1+ ( 2 -x ) ( 2 -x ) d x On adding Eqs. (i) and (ii), we get aligned 2 I & = ₀^ 2 x+ x 1+ x x 2 2 d x & = ₀^ 2 2 ( x+ x) 2+2 x x & = ₀^ 2 2 ( x+ x) 3-( x- x)^2 d x aligned Let x- x=t ( x+ x) d x=d t Now, changing the limit aligned & x 0 t -1 & x / 2 t 1 & 2 I= _ -1 ^1 2 d t 3-t^2 & [ _ -a ^a f(x) d x=2 ₀^a f(x) d x if f(x) is even function ] & 2 I=2 ₀^1 2 d t ( 3 )^2-t^2 & I= 2 [ ⁻¹ ( t 3 ) ]₀^1 & I= 2 ⁻¹ ( 1 3 ) . aligned