Which of the following is the correct integration by parts formula?

• A. ∫ u dv = uv + ∫ v du
• B. ∫ u dv = uv + ∫ u dv
• C. ∫ u dv = -uv + ∫ v du
• D. ∫ u dv = uv - ∫ v du

Answer: (D)

Integration by parts comes from the product rule for differentiation: d(uv)/dx = u(dv/dx) + v(du/dx). If you integrate both sides with respect to x, the left side becomes uv, and you get uv = ∫ u dv + ∫ v du. Solving for the target integral gives ∫ u dv = uv − ∫ v du. This minus sign is essential; it balances the relationship that arises from differentiating a product.

For example, if you take ∫ x e^x dx and choose u = x and dv = e^x dx, then du = dx and v = e^x. The formula yields ∫ x e^x dx = x e^x − ∫ e^x dx = x e^x − e^x + C, which is correct. Using a plus sign in place of the minus would lead to a result that doesn’t match the original integral when differentiated back, so the minus is required.