# AP Calculus AB : Asymptotic and Unbounded Behavior

## Example Questions

### Example Question #3 : Finding Definite Integrals

Explanation:

Remember the fundamental theorem of calculus!

Since our , we can't use the power rule. Instead we end up with:

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 #31 : Finding Integrals

Explanation:

The integral of  is .  The constant 3 is simply multiplied by the integral.

### Example Question #31 : Integrals

Explanation:

To integrate , we need to get the two equations in terms of each other. We are going to use "u-substitution" to create a new variable, , which will equal .

Now, if , then

Multiply both sides by  to get the more familiar:

Note that our , and our original equation was asking for a positive .

That means if we want  in terms of , it looks like this:

Bring the negative sign to the outside:

.

We can use the power rule to find the integral of :

Since we said that , we can plug that back into the equation to get our answer:

### Example Question #61 : Asymptotic And Unbounded Behavior

Evaluate the integral below:

1

Explanation:

In this case we have a rational function as , where

and

can be written as a product of linear factors:

It is assumed that A and B are certain constants to be evaluated. Denominators can be cleared by multiplying both sides by (x - 4)(x + 4). So we get:

First we substitute x = -4 into the produced equation:

Then we substitute x = 4 into the equation:

Thus:

Hence:

### Example Question #1 : Finding Integrals By Substitution

Determine the indefinite integral:

Explanation:

, so this can be rewritten as

Set . Then

and

Substitute:

The outer factor can be absorbed into the constant, and we can substitute back:

### Example Question #1 : Finding Integrals By Substitution

Evaluate:

Explanation:

Set . Then

and

Also, since , the limits of integration change to  and .

Substitute:

### Example Question #1 : Finding Definite Integrals

Explanation:

Remember the fundamental theorem of calculus!

Since our , we can use the reverse power rule to find the indefinite integral or anti-derivative of our function:

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 #5 : Finding Definite Integrals

Explanation:

Remember the fundamental theorem of calculus!

As it turns out, since our , the power rule really doesn't help us.  has a special anti derivative: .

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 #6 : Finding Definite Integrals

Explanation:

Remember the fundamental theorem of calculus!

As it turns out, since our , the power rule really doesn't help us.  is the only function that is it's OWN anti-derivative. That means we're still going to be working with .

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:

Because  is so small in comparison to the value we got for , our answer will end up being

### Example Question #62 : Calculus Ii — Integrals

What is the indefinite integral of ?