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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