A useful technique for evaluating integrals is Integration by Parts: It can be derived from the product formula for the derivatives. At times it is more convenient to express this formula using differentials.
In calculus, and generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into the other integrals. Referred to as the "anti-product rule", the rule arises from product rule of differentiation.
The original integral contains derivative of g; in order to be able to apply the rule, the antiderivative g should be found, and then the resulting integral ∫g f ′ dx should be evaluated.
One can formulate a discrete analogue for sequences, called as summation by parts.
This formula is valid whenever f is differentiable continuously and g is continuous.
Integration by parts is a heuristic rather than a totally mechanical process for solving integrals; given a single function to integrate, the typical strategy is to separate carefully it into a product of two functions ƒ(x)g(x) so that the integral produced by the integration by parts formula is easier to evaluate than original one.
Note that on the right-hand side, ƒ is differentiated and g can be integrated; consequently it is useful to select ƒ as a function which simplifies when differentiated, and/or to choose g as a function which simplifies when integrated.
Alternatively, we may choose ƒ and g so that the product simplifies because of cancellation.
In some applications, it may not be significant to ensure that the integral produced by integration by parts has a simple form; for instance, in numerical analysis, it can suffice that it has small magnitude and so contributes a small error term. In order to apply the method of integration by parts, you should 1st choose your parts. This means choosing u and v. There are always at least 2 ways to choose them, but only one of those ways leads to useful results. You will have to rely on trial and error to determine which of the ways to choose is right one.