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integration by parts

Integration by Parts

A formula used to integrate the product of two functions.

 

Formula:
  
Example 1:Evaluate .
 

Use u = x and dv = ex/2 dx. Then we get du = dx and v = 2ex/2. This can be summarized:

u = xdv = ex/2 dx
du = dxv = 2ex/2

It follows that

 

Example 2:Evaluate .
 

 

Use the following:

u = tan-1xdv = dx
v = x

Thus

 

Example 3:Evaluate .
 

 

Let I =. Proceed as follows:

u = sinxdv = exdx
du = cosx dxv = ex

Thus

Now use integration by parts on the remaining integral . Use the following assignments:

u = cosxdv = exdx
du = –sinx dxv = ex

Thus

Note that appears on both sides of this equation. Replace it with I and then solve.

We finally obtain

 

See also

Integration methods

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