_ - 2 ^ 2 x- ^3 x d x is equal to

Step-by-step solution

The integral is I = _ - /2 ^ /2 x - ^3 x dx . Since x - ^3 x = x(1 - ^2 x) = x ^2 x , the integrand becomes x ^2 x = | x| x . The integral is I = _ - /2 ^ /2 | x| x dx . Since the integrand f(x) = | x| x is an even function because f(-x) = | (-x)| (-x) = |- x| x = | x| x = f(x) , we can write I = 2 ₀^ /2 x x dx . Let u = x , then du = - x dx . When x = 0 , u = 1 . When x = /2 , u = 0 . I = 2 ₁⁰ u (-du) = 2 ₀¹ u^ 1/2 du . I = 2 [ u^ 3/2 3/2 ]₀¹ = 2 2 3 [u^ 3/2 ]₀¹ = 4 3 (1 - 0) = 4 3 . Answer: 4 3