The value of ₀^ | ^3 x | d x is

Step-by-step solution

To evaluate ₀^ | ^3 x | dx , note that x 0 for x [0, ] , so the absolute value may be removed without changing the integral. The integral becomes ₀^ ^3 x dx , which is separable by writing ^3 x = x (1 - ^2 x) using the identity ^2 x = 1 - ^2 x . Substituting u = x , du = - x dx , with limits u=1 when x=0 and u=-1 when x= , the integral becomes ₁⁻¹ (1 - u^2)(- du) = _ -1 ^1 (1 - u^2) du . Exploiting the evenness of 1 - u^2 on the symmetric interval [-1,1] , the integral is 2 ₀^1 (1 - u^2) du = 2 [u - u^3 3 ]₀^1 = 2(