Integration by Parts

A very useful technique for evaluating integrals is Integration by Parts: It is derived from the product formula for derivatives. Sometimes it is more convenient to express this formula using differentials.

In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other integrals. Often referred to as the "anti-product rule", the rule arises from the product rule of differentiation.

The original integral contains the derivative of g; in order to be able to apply the rule, the antiderivative g must be found, and then the resulting integral integration of g f' dx must be evaluated.

One can also formulate a discrete analogue for sequences, called summation by parts. This formula is valid whenever f is continuously differentiable and g is continuous.

Integration by parts is a heuristic rather than a purely mechanical process for solving integrals; given a single function to integrate, the typical strategy is to carefully separate it into a product of two functions f(x)g(x) such that the integral produced by the integration by parts formula is easier to evaluate than the original one.

Note that on the right-hand side, f is differentiated and g is integrated; consequently it is useful to choose f as a function that simplifies when differentiated, and/or to choose g as a function that simplifies when integrated.

Alternatively, we may choose f and g such that the product simplifies due to cancellation. In some applications, it may not be necessary to ensure that the integral produced by integration by parts has a simple form; for example, in numerical analysis, it may suffice that it has small magnitude and so contributes only a small error term. In order to apply the method of integration by parts, you must first choose your parts. This means choosing u and v. There are always at least two ways to choose them, but usually only one of those ways leads to useful results. Often you will have to rely on trial and error to determine which of the ways to choose is the right one.