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Power rule, Product rule, Quotient rule, Chain rule
Lists of integrals
Improper integrals
Integration by:
parts, disks, cylindrical
shells, substitution,
trigonometric substitution,
partial fractions, changing order
In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other (ideally simpler) integrals. The rule arises from the product rule of differentiation.
If u = f(x), v = g(x), and the differentials du = f '(x) dx and dv = g'(x) dx, then integration by parts states that
i.e.
Contents
Rule
Suppose f(x) and g(x) are two continuously differentiable functions. The product rule states
Integrating both sides gives
Rearranging terms
From the above one can derive the integration by parts rule, which states that, given an interval with endpoints a and b,
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