Consider the integral I_m= ₀^ 2 m x x d x , where m is a positive integer. What is I₂+I₃ equal to?

Step-by-step solution

aligned & I₂= ₀^ 4 x x d x= ₀^ 2 2 x 2 x x d x & =2 ₀^ 2 x x 2 x x dx & =4 ₀^ x (1-2 ^2 x ) d x & =4 ₀^ x d x-8 ₀^ x ^2 x d x=0 aligned aligned & Let x = t & xdx = dt & =4( - 0)-0=0 & I₃= ₀^ 6 x x d x & = ₀^ 2 [3 x-4 ^3 x ] 3 x x d x aligned aligned & = ₀^ 2 (3-4 ^2 x ) 3 x d x & = ₀^ 6 3 x d x- ₀^ 8 ^2 x 3 x d x & = 6 3 x 3 ₀^ -8 1 2 [ ₀^ x( 4 x- 2 x) d x ] & =6- 8 2 [ ₀^ ( x 4 x- x 2 x) d x ] aligned aligned & = -8 2 [ 1 2 ₀^ 3 x- 5 x- x+ 3 x d x ] & =-2 [ 2 3 x 3 - 5 x 5 - x ]₀^ =0 aligned Hence, I ₂+ I ₃=0+0=0