All AP Calculus BC Resources
Example Questions
Example Question #11 : Fundamental Theorem Of Calculus And Techniques Of Antidifferentiation
Evaluate:
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 #12 : Fundamental Theorem Of Calculus And Techniques Of Antidifferentiation
Evaluate:
The integral does not converge
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 #1 : Improper Integrals
Evaluate:
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 #1 : Improper Integrals
Evaluate .
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,
.