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

It follows that

Example 2:Evaluate .

Use the following:

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

Thus

Example 3:Evaluate .

Let I =. Proceed as follows:

 u = sinx dv = exdx du = cosx dx v = ex

Thus

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

 u = cosx dv = exdx du = –sinx dx v = ex

Thus

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

We finally obtain