In calculus, and more generally in mathematical analysis, integration by parts 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 find the antiderivative of a product of functions into an ideally simpler antiderivative. The rule can be derived in one line by simply integrating the product rule of differentiation.
If u = u(x), v = v(x), and the differentials du = u '(x) dx 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 integral and Lebesgue–Stieltjes integral. One can also formulate a discrete analogue for sequences, called summation by parts.
Read more about Integration By Parts: Visualisation, Recursive Integration By Parts, Higher Dimensions, Infinite Congruence Theorem
Famous quotes containing the words integration and/or parts:
“The only phenomenon with which writing has always been concomitant is the creation of cities and empires, that is the integration of large numbers of individuals into a political system, and their grading into castes or classes.... It seems to have favored the exploitation of human beings rather than their enlightenment.”
—Claude Lévi-Strauss (b. 1908)
“Language is a living thing. We can feel it changing. Parts of it become old: they drop off and are forgotten. New pieces bud out, spread into leaves, and become big branches, proliferating.”
—Gilbert Highet (1906–1978)