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 more specific idea of evolution now reached is—a change from an indefinite, incoherent homogeneity to a definite, coherent heterogeneity, accompanying the dissipation of motion and integration of matter.”
—Herbert Spencer (1820–1903)
“Water, earth, air, fire, and the other parts of this structure of mine are no more instruments of your life than instruments of your death. Why do you fear your last day? It contributes no more to your death than each of the others. The last step does not cause the fatigue, but reveals it. All days travel toward death, the last one reaches it.”
—Michel de Montaigne (1533–1592)