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  
\[\frac{dv}{dx}=x^n\]
.

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