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 transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found.

The rule can be derived in one line simply by 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

$\int~u(x).v'(x)\,dx~=~u(x).v(x)~-~\int~u'(x).v(x)\,dx$

or more compactly:

$\int~u.dv~=~uv~-~\int~v.du$

Again, using alternate notation:

$\int~u(x).\frac{dv}{dx}.dx~=~u(x).v(x)~-~\int~v(x).\frac{du}{dx}.dx$
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