Integration by parts

Mathematician Brook Taylor discovered integration by parts, first publishing the idea in 1715.

The discrete analogue for sequences is called summation by parts.

This is to be understood as an equality of functions with an unspecified constant added to each side.

and applying the fundamental theorem of calculus gives the definite integral version:

contains the derivative v'; to apply the theorem, one must find v, the antiderivative of v', then evaluate the resulting integral

[citation needed] Integrating the product rule for three multiplied functions,

Assuming that the curve is locally one-to-one and integrable, we can define

Alternatively, one may choose u and v such that the product u′ (∫v dx) simplifies due to cancellation.

using the chain rule and v integrates to tan x; so the formula gives:

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.

An example commonly used to examine the workings of integration by parts is

A similar method is used to find the integral of secant cubed.

Two other well-known examples are when integration by parts is applied to a function expressed as a product of 1 and itself.

using a combination of the inverse chain rule method and the natural logarithm integral condition.

Also, in some cases, polynomial terms need to be split in non-trivial ways.

Integration by parts is often used as a tool to prove theorems in mathematical analysis.

Integration by parts illustrates it to be an extension of the factorial function:

Integration by parts is often used in harmonic analysis, particularly Fourier analysis, to show that quickly oscillating integrals with sufficiently smooth integrands decay quickly.

A similar method can be used to find the Laplace transform of a derivative of a function.

satisfies these conditions then its Fourier transform decays at infinity at least as quickly as 1/|ξ|k.

Using the same idea on the equality stated at the start of this subsection gives

Extending this concept of repeated partial integration to derivatives of degree n leads to

are readily available (e.g., plain exponentials or sine and cosine, as in Laplace or Fourier transforms), and when the nth derivative of

The latter condition stops the repeating of partial integration, because the RHS-integral vanishes.

The essential process of the above formula can be summarized in a table; the resulting method is called "tabular integration"[5] and was featured in the film Stand and Deliver (1988).

This process comes to a natural halt, when the product, which yields the integral, is zero (i = 4 in the example).

In this case the repetition may also be terminated with this index i.This can happen, expectably, with exponentials and trigonometric functions.

In this case the product of the terms in columns A and B with the appropriate sign for index i = 2 yields the negative of the original integrand (compare rows i = 0 and i = 2).

Integration by parts can be extended to functions of several variables by applying a version of the fundamental theorem of calculus to an appropriate product rule.

is the outward unit normal vector to the boundary, integrated with respect to its standard Riemannian volume form