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Integration by Parts Formula

Integration by parts or partial integration is a theorem that relates the integral of a product of functions to the integral of their derivative and antiderivative.

The Integration by parts formula is :

\[\large \int u\;v\;dx=u\int v\;dx-\int\left(\frac{du}{dx}\int v\;dx\right)dx\]

Where $u$ and $v$ are the differentiable functions of $x$

Product of two functions

The theorem can be derived as follows. Suppose $u(x)$ and $v(x)$ are two continuously differentiable functions. The product rule states:

\[\large \frac{d}{dx}\left(u(x)v(x)\right)=v(x)\frac{d}{dx}(u(x))+u(x)\frac{d}{dx}(v(x))\]

Integrating both sides with respect to x,

\[\large \int \frac{d}{dx}(u(x)v(x))dx=\int u'(x)v(x)dx+\int u(x)v'(x)dx\]

then applying the definition of indefinite integral,

\[\large u(x)v(x)=\int u'(x)v(x)dx+\int u(x)v'(x)dx\]

\[\large \int u(x)v'(x)dx=u(x)v(x)-\int u'(x)v(x)dx\]

Gives the formula for integration by parts. Since $du$ and $dv$ are differentials of a function of one variable $x$,

\[\large du=u'(x)dx\;\;dv=v'(x)dx\]

\[\large \int u(x)dv=u(x)v(x)-\int v(x)du\]

The original integral $\int uv’\;dx$ contains $v’$ (derivative of $v$); in order to apply the theorem, $v$ (antiderivative of $v’$) must be found, and then the resulting integral $\int vu’\; dx$ must be evaluated.