AP Calculus BC : Improper Integrals

Study concepts, example questions & explanations for AP Calculus BC

varsity tutors app store varsity tutors android store

Example Questions

Example Question #1 : Improper Integrals

Evaluate:

Possible Answers:

Correct answer:

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 #2 : Improper Integrals

Evaluate:

Possible Answers:

The integral does not converge

Correct answer:

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 #3 : Improper Integrals

Evaluate:

Possible Answers:

Correct answer:

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 #4 : Improper Integrals

Evaluate .

Possible Answers:

Correct answer:

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, 

.

 

Learning Tools by Varsity Tutors

Incompatible Browser

Please upgrade or download one of the following browsers to use Instant Tutoring: