# AP Calculus AB : Asymptotic and Unbounded Behavior

## Example Questions

### Example Question #24 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate the integral:

1

Explanation:

In order to find the antiderivative, add 1 to the exponent and divide by the exponent.

Evaluate:

Explanation:

### Example Question #26 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate:

Explanation:

You should first know that the derivative of .

Therefore, looking at the equation you can see that the antiderivative should involve something close to:

Now to figure out what value represents the square take the derivative of  and set it equal to what the original integral contained.

Since the derivative of  contains a 3 that the integral does not show, we know that the square is equal to . Thus, the answer is .

### Example Question #31 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate:

Explanation:

The antiderivative of . The derivative of . However, since there is no 2 in the original integral, we must divide  by 2. Therefore, the answer is

### Example Question #32 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate the integral:

Explanation:

When taking the antiderivative add one to the exponent and then divide by the exponent.

### Example Question #51 : Asymptotic And Unbounded Behavior

Evaluate the integral:

Cannot be evaluated

Explanation:

The derivative of . Therefore, the antiderivative of  is equal to itself.

### Example Question #34 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate:

Can't be determined from the information given.

Explanation:

and

Recall that  is an odd function and  is an even function.

Thus, since  is an odd function, the integral of this function from  to  will be zero.

### Example Question #35 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate this indefinite integral:

Explanation:

To approach this problem, first rewrite the integral expression as shown below:

.

Then, recognize that , and substitute this into the integral expression:

Use substitution, letting  and .  The integral can then be rewritten as

Evaluating this integral gives

.

Finally, substituting  back into this expression gives the final answer:

(As this is an indefinite integral,  must be included).

Evaluate:

Explanation:

Find