Integration by parts

In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a theorem that relates the integral of a product of functions to the integral of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be derived in one line simply by integrating the product rule of differentiation.

If u = u(x) and du = u′(x) dx, while v = v(x) and dv = v′(x) dx, then integration by parts states that:

or more compactly:

More general formulations of integration by parts exist for the Riemann–Stieltjes and Lebesgue–Stieltjes integrals. The discrete analogue for sequences is called summation by parts.

The theorem can be derived as follows. Suppose u(x) and v(x) are two continuously differentiable functions. The product rule states (in Leibniz's notation):

Integrating both sides with respect to x,

then applying the definition of indefinite integral,

gives the formula for integration by parts.

Since du and dv are differentials of a function of one variable x,

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