# AP Calculus AB : Asymptotic and Unbounded Behavior

## Example Questions

### Example Question #81 : Asymptotic And Unbounded Behavior

Explanation:

Remember the fundamental theorem of calculus!

Since our , we can't use the power rule, as it has a special antiderivative:

Remember to include the  for any anti-derivative or integral taken!

Now we can plug that equation into our FToC equation:

Notice that the c's cancel out. Plug in the given values for a and b and solve:

### Example Question #32 : Calculus Ii — Integrals

Explanation:

Remember the fundamental theorem of calculus!

Since our , we can't use the power rule, as it has a special antiderivative:

Remember to include the  for any anti-derivative or integral taken!

Now we can plug that equation into our FToC equation:

Notice that the c's cancel out. Plug in the given values for a and b and solve:

### Example Question #68 : Calculus Ii — Integrals

What is the indefinite integral of ?

Undefined

Explanation:

To find the indefinite integral of our equation, we can use the reverse power rule.

To use the reverse power rule, we raise the exponent of the  by one and then divide by that new exponent.

Remember that, when taking the integral, we treat constants as that number times , since anything to the zero power is . Treat  as .

When taking an integral, be sure to include a  stands for "constant". Since taking the derivative of a constant whole number will always equal , we include the  to anticipate the possiblity of the equation actually being  or  instead of just  .

### Example Question #69 : Calculus Ii — Integrals

What is the indefinite integral of ?

Explanation:

is a special function.

The indefinite integral is .

Even though it is a special function, we still need to include a  stands for "constant". Since taking the derivative of a constant whole number will always equal , we include the  to anticipate the possiblity of the equation actually being  or  instead of just  .

### Example Question #70 : Calculus Ii — Integrals

What is the indefinite integral of ?

Explanation:

To solve this problem, we can use the anti-power rule or reverse power rule. We raise the exponent on the variables by one and divide by the new exponent.

For this problem, we'll treat  as  since anything to the zero power is one.

Since the derivative of any constant is , when we take the indefinite integral, we add a  to compensate for any constant that might be there.

From here we can simplify.

### Example Question #291 : Ap Calculus Ab

Explanation:

Remember the fundamental theorem of calculus!

Since our , we can't use the power rule. Instead we must use u-substituion.  If

Remember to include the  for any anti-derivative or integral taken!

Now we can plug that equation into our FToC equation:

Notice that the c's cancel out. Plug in the given values for a and b and solve:

### Example Question #34 : Calculus Ii — Integrals

Explanation:

Remember the fundamental theorem of calculus!

Since our , we can use the reverse power rule to find that the antiderivative is:

Remember to include a  for any integral or antiderivative taken!

Now we can plug that equation into our FToC equation:

Notice that the c's cancel out. Plug in the given values for a and b and solve:

### Example Question #35 : Calculus Ii — Integrals

If n is a positive integer, find .

0

Explanation:

We can find the integral using integration by parts, which is written as follows:

Let and . We can get the box below:

Now we can write:

### Example Question #71 : Calculus Ii — Integrals

What is the indefinite integral of ?

Explanation:

To solve this problem, we can use the anti-power rule or reverse power rule. We raise the exponent on the variables by one and divide by the new exponent.

For this problem, we'll treat  as  since anything to the zero power is one.

Since the derivative of any constant is , when we take the indefinite integral, we add a  to compensate for any constant that might be there.

From here we can simplify.

### Example Question #72 : Calculus Ii — Integrals

What is the indefinite integral of ?