Parts

In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, hopefully simpler, integrals. The rule arises from the product rule of differentiation. more...

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The rule

Suppose f(x) and g(x) are two continuously differentiable functions. Then the integration by parts rule states that given an interval with endpoints a, b, one has

where we use the common notation

The rule is shown to be true by using the product rule for derivatives and the fundamental theorem of calculus. Thus

In the traditional calculus curriculum, this rule is often stated using indefinite integrals in the form

or in an even shorter form, if we let u = f(x), v = g(x) and the differentials du = f ′(x) dx and dv = g′(x) dx, then it is in the form in which it is most often seen:

Note that 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 ∫g f′ dx must be evaluated.

One can also formulate a discrete analogue for sequences, called summation by parts.

An alternative notation has the advantage that the factors of the original expression are identified as f and g, but the drawback of a nested integral:

This formula is valid whenever f is continuously differentiable and g is continuous.

More general formulations of integration by parts exist for the Riemann-Stieltjes integral and Lebesgue-Stieltjes integral.

Note: More complicated forms such as the one below are also valid:

Strategy

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. The following form is useful in illustrating the best strategy to take:

Read more at Wikipedia.org

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