Integration by parts

Let $a<b$ . Suppose $u,v:[a,b]\to\mathbb R$ are continuous on $[a,b]$ , differentiable on $(a,b)$ , and their derivatives extend to Riemann-integrable functions on $[a,b]$ . Then

\int_a^b u(x)\,v'(x)\,dx = u(b)v(b)-u(a)v(a) - \int_a^b u'(x)\,v(x)\,dx.

Remarks

This is the integral form of the product rule: apply the fundamental theorem of calculus to $(uv)'=u'v+uv'$ .