Compute ∫ x e^x dx by parts; which expression represents its antiderivative?

Compute ∫ x e^x dx by parts; which expression represents its antiderivative?

• A. e^x (x + 1) + C
• B. e^x x + C
• C. e^x / x + C
• D. e^x (x - 1) + C

Answer: (D)

Integration by parts helps when you integrate a product like x e^x. Use ∫ u dv = uv − ∫ v du. Let u = x and dv = e^x dx, so du = dx and v = e^x. Then ∫ x e^x dx = x e^x − ∫ e^x dx = x e^x − e^x + C = e^x(x − 1) + C. A quick check: differentiate e^x(x − 1) using the product rule to get e^x(x − 1) + e^x = e^x x, which matches the original integrand. So the antiderivative is e^x(x − 1) + C.

Integration by parts helps when you integrate a product like x e^x. Use ∫ u dv = uv − ∫ v du. Let u = x and dv = e^x dx, so du = dx and v = e^x. Then

∫ x e^x dx = x e^x − ∫ e^x dx = x e^x − e^x + C = e^x(x − 1) + C.

A quick check: differentiate e^x(x − 1) using the product rule to get e^x(x − 1) + e^x = e^x x, which matches the original integrand.

So the antiderivative is e^x(x − 1) + C.