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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 = x dv = ex/2 dx
du = dx v = 2ex/2

It follows that

 

Example 2: Evaluate .
 

 

Use the following:

u = tan-1 x dv = dx
v = x

Thus

 

Example 3: Evaluate .
 

 

Let I =. Proceed as follows:

u = sin x dv = ex dx
du = cos x dx v = ex

Thus

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

u = cos x dv = ex dx
du = –sin x dx v = 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

 


  this page updated 21-feb-16
Mathwords: Terms and Formulas from Algebra I to Calculus
written, illustrated, and webmastered by Bruce Simmons
NCTM Web Bytes December 2004 Web Bytes March 2005 Web Bytes