∫ 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).