Basic Analysis I — Real Analysis · Problem 5.3.5

The Riemann Integral — Jiri Lebl

Mathematics — Real Analysis · integration-by-parts

Problem

Prove integration by parts. That is, suppose FF and GG are continuously differentiable functions on [a,b][a,b]. Then prove ∫abF(x)G′(x) dx=F(b)G(b)−F(a)G(a)−∫abF′(x)G(x) dx.\int_a^b F(x)G'(x)\,dx = F(b)G(b) - F(a)G(a) - \int_a^b F'(x)G(x)\,dx.
Intuitive explanation

The key insight is to recognize that the product rule for derivatives—(FG)′=F′G+FG′(FG)' = F'G + FG'—already encodes the relationship you need. Integrate both sides of this identity over [a,b][a,b], and the left side becomes a simple evaluation by the Fundamental Theorem of Calculus, while the right side splits into two integrals. Rearranging isolates the integral you want in terms of the other pieces.

Intuitive explanation

The key insight is to recognize that the product rule for derivatives—(FG)′=F′G+FG′(FG)' = F'G + FG'—already encodes the relationship you need. Integrate both sides of this identity over [a,b][a,b], and the left side becomes a simple evaluation by the Fundamental Theorem of Calculus, while the right side splits into two integrals. Rearranging isolates the integral you want in terms of the other pieces.

Related theory

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