# GRE Subject Test: Math : Integration by Parts

## Example Questions

### Example Question #2 : Integrals

Integrate the following.

Explanation:

Integration by parts follows the formula:

So, our substitutions will be  and

which means  and

Plugging our substitutions into the formula gives us:

Since  , we have:

, or

### Example Question #3 : Integrals

Evaluate the following integral.

Explanation:

Integration by parts follows the formula:

In this problem we have  so we'll assign our substitutions:

and

which means  and

Including our substitutions into the formula gives us:

We can pull out the fraction from the integral in the second part:

Completing the integration gives us:

### Example Question #4 : Integrals

Evaluate the following integral.

Explanation:

Integration by parts follows the formula:

Our substitutions will be  and

which means  and .

Plugging our substitutions into the formula gives us:

Look at the integral: we can pull out the  and simplify the remaining  as

.

We now solve the integral:  , so:

### Example Question #5 : Integrals

Evaluate the following integral.

Explanation:

Integration by parts follows the formula:

.

Our substitutions are  and

which means  and .

Plugging in our substitutions into the formula gives us

We can pull   outside of the integral.

Since , we have