Choose the correct option regarding the following definite integrals (i) ₀^ / 2 ^m(x) (x) d x= 1 m+1 (ii) ₀^ / 2 (x) ^n(x) d x= 1 n+1

Step-by-step solution

(i) ₀^ / 2 ^m x x d x Put, x=t aligned x d x & =d t= ₀^ / 2 t^m d t & = [ t^ m+1 m+1 ]₀^ / 2 = [ ( x)^ m+1 m+1 ]₀^ / 2 aligned aligned & = 1 m+1 [( x)^ m+1 ]₀^ / 2 & = 1 m+1 [ ^ m+1 ( 2 )- ^ m+1 (0) ] & = 1 m+1 [(1)-(0)]= 1 m+1 aligned (ii) ₀^ / 2 x ^n(x)=d x aligned x & =t - x d x & =d t x d x & =-d t ₀^ / 2 t^n (-d t) & =- ₀^ / 2 t^n d t = & - [ t^ n+1 n+1 ]₀^ / 2 =- [ x^ n+1 n+1 ]₀^ / 2 & = -1 n+1 [ ^ n+1 x ]₀^ / 2 aligned aligned & = -1 n+1 [ ^ n+1 2 - ^ n+1 (0) ] & = -1 n+1 [0-1]= 1 n+1 aligned Hence, option (