PlanetMath: integration by parts

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integration by parts

(Theorem)

When we want to integrate a product of two functions, it is sometimes preferable to simplify the integrand by integrating one of the functions and differentiating the other. This process is called integrating by parts, and is done in the following way, where and are functions of .

$\displaystyle \int u\cdot v'\; dx = u\cdot v - \int v\cdot u'\; dx$

This process may be repeated indefinitely, and in some cases it may be used to solve for the original integral algebraically. For definite integrals, the rule appears as

$\displaystyle \int_a^b u(x)\cdot v'(x)\; dx = (u(b)\cdot v(b)-u(a)\cdot v(a)) - \int_a^b v(x)\cdot u'(x)\; dx$

Proof: Integration by parts is simply the antiderivative of a product rule. Let $G(x)=u(x)\cdot v(x)$ . Then,

$\displaystyle G'(x) = u'(x)v(x) + u(x)v'(x)$

Therefore,

$\displaystyle G'(x) - v(x)u'(x) = u(x)v'(x)$

We can now integrate both sides with respect to

to get

$\displaystyle G(x) - \int v(x)u'(x)\; dx = \int u(x)v'(x)\; dx$

which is just integration by parts rearranged.
Example: We integrate the function $f(x)=x\sin x$ : Therefore we define

and $v'(x)=\sin x$ . So integration by parts yields us:

$\displaystyle \int x\sin x\mathit{dx}=-x\cos x+\int\cos x\mathit{dx}=-x\cos x+\sin x+C,$

where

is an arbitrary constant.

"integration by parts" is owned by mathwizard. [ full author list (2) | owner history (1) ]

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See Also: general formulas for integration, integration of $\sqrt{x^2+1}$

Attachments:: a lecture on integration by parts (Feature) by alozano
example of integration by parts involving algebraic manipulation (Example) by Wkbj79

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Cross-references: sides, product rule, antiderivative, definite integrals, integral, functions, product
There are 13 references to this entry.

This is version 8 of integration by parts, born on 2002-02-25, modified 2005-12-23.
Object id is 2683, canonical name is IntegrationByParts.
Accessed 8127 times total.

Classification:

AMS MSC:

26A36 (Real functions :: Functions of one variable :: Antidifferentiation)

Pending Errata and Addenda

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