Integration by parts: the product rule run backwards, first published in 1715
Integrating a product of two functions is often hard, but trading it for a different product can make the problem easy. Integration by parts does exactly that swap, and it falls straight out of the product rule for derivatives. Brook Taylor published the idea in 1715.
Start with two continuously differentiable functions, u and v. The product rule says the derivative of uv equals u-prime times v plus u times v-prime. Integrate both sides, remember that integrating a derivative just gives back the original function, and rearrange: the integral of u times v-prime equals uv minus the integral of u-prime times v. In compact shorthand, the integral of u dv is uv minus the integral of v du, with an unspecified constant understood on each side.
For a definite integral from a to b, the fundamental theorem of calculus turns the uv term into a boundary contribution, u(b)v(b) minus u(a)v(a), followed by the leftover integral. The indefinite and definite forms agree once limits are applied, though they are not interchangeable as written. To use the rule you must first find v, an antiderivative of the factor you labelled v-prime, and then hope the new integral is friendlier than the old one.
The smoothness assumptions can be relaxed. The formula still holds when u is absolutely continuous and v-prime is merely Lebesgue integrable, not necessarily continuous; if v-prime jumps somewhere, its antiderivative may lack a derivative at that point. On unbounded intervals even those conditions can fail while the formula survives, provided the boundary term and remaining integral are finite. Choosing u as e to the x over x squared with v-prime as e to the minus x on the interval from 1 to infinity is one such case; another pairs e to the minus x with sine of x divided by x.
Broader versions exist for Riemann–Stieltjes and Lebesgue–Stieltjes integrals, and there is a variant for functions of bounded variation. For sequences rather than functions, the same trick has a discrete counterpart known as summation by parts.
Source: Integration by parts
