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 \[\frac{dv}{dx}=x^n\]
.