Origin and wording of the formula

Why a new rule?

The derivative of a product is (uv)' = u'v + uv'. By integrating both sides, we obtain an identity relating the integral of u'v to the integral of uv'. This is the basis of integration by parts (IPP), an indispensable tool when a direct antiderivative does not exist but the product can be derived or integrated in stages.

The formula

Undefined version: ∫(u(x) v'(x) dx) = u(x) v(x) − ∫(u'(x) v(x) dx)

Defined version (limits a, b): ∫_a^b u v' dx = [u v]_a^b – ∫_a^b u' v dx

Quick proof

We start with (uv)’ = u’ v + u v’. Integrating over [a, b]: ∫_a^b (uv)’ dx = ∫_a^b u’ v dx + ∫_a^b u v’ dx

However, ∫_a^b (uv)' dx = [u v]_a^b (Fundamental Theorem). Hence the formula, isolating ∫_a^b u v'.

Immediate example

Calculate I = ∫_a^b x * e^x dx. Let u = x, dv = e^x dx, so du = dx, v = e^x. I = x e^x – ∫ e^x dx = x e^x – e^x + C = (x–1) e^x + C.

Summary table

Element	Role	Must be
u	derivative factor	simple to derive
dv	integrating factor	simple to integrate
du	derivative of u	simpler than u
v	antiderivative of dv	calculable explicitly

This rule transforms a complicated product into another, generally simpler, product, provided that u and dv are chosen carefully.