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

.