Integration by parts

An identity relating the integral of a product to boundary terms and another integral.

Integration by parts

Integration by parts: Let $a<b$ and let $u,v:[a,b]\to\mathbb{R}$ be functions that are differentiable on $[a,b]$ , with $u'$ and $v'$ Riemann integrable 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.

This is the integral form of the product rule from differentiation rules , typically justified using fundamental theorem of calculus II .