$\displaystyle \int udv$ $\,=\,$ $uv$ $-$ $\displaystyle \int vdu$
A mathematical approach of finding the integral of a product of functions from the integral of their derivative and antiderivative, is called the integration by parts.
$f{(x)}$ and $g{(x)}$ are two functions. Take $u = f{(x)}$ and $v = g{(x)}$
The derivatives of both functions in differential form are written in the following mathematical form.
$(1) \,\,\,$ $du = f'{(x)}dx$
$(2) \,\,\,$ $dv = g'{(x)}dx$
Let’s study each part of this integral property.
$\displaystyle \int udv \,=\, \int f{(x)}.g'{(x)} dx$ and $\displaystyle \int vdu \,=\, \int g{(x)}.f'{(x)} dx$
Each equation represents the integration of a product of a function and derivative of another function, and $uv$ is a product of a function and an antiderivative of another function.
Learn how to derive the integration by parts formula in mathematical form.
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