### All AP Calculus AB Resources

## Example Questions

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

Evaluate the integral:

**Possible Answers:**

1

**Correct answer:**

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

### Example Question #25 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate:

**Possible Answers:**

**Correct answer:**

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

Evaluate:

**Possible Answers:**

**Correct answer:**

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:

**Possible Answers:**

**Correct answer:**

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:

**Possible Answers:**

**Correct answer:**

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:

**Possible Answers:**

Cannot be evaluated

**Correct answer:**

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:

**Possible Answers:**

Can't be determined from the information given.

**Correct answer:**

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:

**Possible Answers:**

**Correct answer:**

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).

### Example Question #1 : Finding Definite Integrals

Evaluate:

**Possible Answers:**

**Correct answer:**

### Example Question #1 : Finding Integrals

Find

**Possible Answers:**

**Correct answer:**

This is most easily solved by recognizing that .

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