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 $u$Â 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.
