What is "integration by parts"

n. (label en analysis) (l en A method of integration directly related to the rule for differentiation of products); can be written as $int\; u\; dv\; =\; uv\; -\; int\; v\; du$.

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 and , while and , then integration by parts states that:

∫u(x)vʹ(x) dx = u(x)v(x) − ∫v(x) uʹ(x)dx

or more compactly:

∫u dv = uv − ∫v du.

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.