## Integration By Parts

Integration by parts is a commonly used technique of integration, used especially to integrate products. It is derived from the product rule for differentiating a product..
The product rule states that for functions
$u, \; v$
,
$\frac{(uv)}{dx}= \frac{du}{dx}v+ u \frac{dv}{dx} \rightarrow uv = \int \frac{du}{dx}v+ u \frac{dv}{dx} \rightarrow \int \frac{dv}{dx}udx= uv- \int\frac{du}{dx}v dx$
.
Example:
$\int xe^x dx$
.
Let
$u=x, \; \frac{dv}{dx}=e^x$
then
$\frac{du}{dx}=1, \; v=e^x$
.
$\int xe^x dx = xe^x - \int e^xdx=xe^x-e^x$
.
When using integration by parts, if one of the factors is a power of
$x$
say
$x^n$
then let
$u=x^n$
except when the other factor is a logarithm, when you set
$\frac{dv}{dx}=x^n$
.