Integration by Parts

Integral Calculus, Integration by Parts

Integration by Parts

Another technique is integration by parts. Let u(x) and v(x) be differentiable on a certain interval. We would like to integrate the function u×v′, but we don't know how. By the product rule, the derivative of uv is uv′+vu′. Thus the integral of uv′ + the integral of vu′ yields uv. Write it this way.

∫ uv′ = uv - ∫ vu′

We may know how to derive the latter integral, thus giving the former.

Assume we want to integrate log(x). (Log is defined in the next section; its derivative is 1/x.) Let u = log(x) and let v = x. Thus log(x) is uv′. By parts, the integral becomes x×log(x) - the integral of (x times the derivative of log(x)). The second integrand reduces to 1, hence the answer is x×log(x)-x. Take the derivative to reproduce log(x).