# AP Calculus BC : Improper Integrals

## Example Questions

### Example Question #1 : Improper Integrals

Evaluate:

Explanation:

First, we will find the indefinite integral using integration by parts.

We will let  and .

Then  and .

To find , we use another integration by parts:

, which means that , and

, which means that, again, .

Since

, or,

for all real , and

,

by the Squeeze Theorem,

.

### Example Question #1 : Improper Integrals

Evaluate:

The integral does not converge

Explanation:

First, we will find the indefinite integral, .

We will let  and .

Then,

and .

and

Now, this expression evaluated at is equal to

.

At it is undefined, because does not exist.

We can use L'Hospital's rule to find its limit as , as follows:

and , so by L'Hospital's rule,

Therefore,

### Example Question #51 : Integrals

Evaluate:

Explanation:

Rewrite the integral as

.

Substitute . Then

and . The lower bound of integration stays , and the upper bound becomes , so

Since , the above is equal to

.

### Example Question #2 : Improper Integrals

Evaluate .

Explanation:

By the Formula Rule, we know that . We therefore know that .

Continuing the calculation:

By the Power Rule for Integrals,  for all  with an arbitrary constant of integration . Therefore:

.

So,

.