₀^ / 2 ^3 x x d x ^4 x+ ^4 x =

Step-by-step solution

= ₀^ / 2 ^3 ( 2 -x ) ( 2 -x ) ^4 ( 2 -x )+ ^4 ( 2 -x ) d x Adding Eqs. (i) and (ii), we get aligned 2 I & = ₀^ / 2 ( ^3 x x+ ^3 x x ) ^4 x+ ^4 x d x & = ₀^ / 2 x x ( ^2 x+ ^2 x ) ^4 x+ ^4 x d x & = ₀^ / 2 x x ^4 x+ ^4 x d x & = ₀^ / 2 x x ^4 x ( ^4 x+1 ) d x & = ₀^ / 2 ^4 ^2 x ( ^4 x+1 ) d x aligned Let ^2 x=t Differentiating w.r.t. x , we get array ll & 2 x ^2 x d x=d t & x ^2 x d x= d t 2 & 2 I= 1 2 ₀^ d t t^2+1 = 1 2 [ ⁻¹ t ]₀^ & 2 I= 1 2 [ ⁻¹( )- ⁻¹(0) ]= 1 2 2 & 2 I= 4 I= 8 array