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In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. The rule arises from the product rule of differentiation.
Suppose f(x) and g(x) are two continuously differentiable functions. Then the integration by parts rule states that for endpoints a, b
where we use the common notation
The rule is shown to be true by using the product rule for derivatives and the fundamental theorem of calculus. Thus
In the traditional calculus curriculum, this rule is often stated using indefinite integrals in the form
or in an even shorter form, if we let u = f(x), v = g(x) and the differentials du = f′(x) dx and dv = g′(x) dx, then it is in the form in which it is most often seen:
One can also formulate a discrete analogue for sequences, called summation by parts.
Note that the original integral contains the derivative of g; in order to be able to apply the rule, you need to find its antiderivative g and then you still have to evaluate the resulting integral of ∫g f ' dx.
An alternative notation has the advantage that the factors of the original expression are identified as f and g, but the drawback of a nested integral:
This formula is valid whenever f is continuously differentiable and g is continuous.
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