Question Detail

To solve ∫ 2x³e^(-x⁴) dx, use u-substitution. Let u = -x⁴. Differentiate u with respect to x: du/dx = -4x³, so du = -4x³ dx. This means x³ dx = -(1/4) du. Substitute these into the integral: ∫ 2e^u (-(1/4)) du = -(2/4) ∫ e^u du = -(1/2) ∫ e^u du. The integral of e^u is e^u. So, -(1/2)e^u + C. Substituting back u = -x⁴, the final answer is -(1/2)e^(-x⁴) + C.