### All Calculus 2 Resources

## Example Questions

### Example Question #21 : Definite Integrals

Evaluate the indefinite integral .

**Possible Answers:**

None of the other answers

**Correct answer:**

None of the other answers

The correct answer is .

The integral itself is not too difficult to take, simply use the Power Rule on the and terms. The trick is to be careful when integrating . is a constant value (about ) not a variable, so it must be integrated accordingly.

### Example Question #22 : Definite Integrals

Evaluate .

**Possible Answers:**

None of the other answers

**Correct answer:**

This integral requires integration by parts followed by u-substitution. Here are the details

. Start

. Factor out the

Set up integration by parts with , . We then have and . Afterward, we use the integration by parts formula .

.

Now at this point we use u-substitution to evaluate the 2nd integral. Let , then and therefore . Substituting into the integral we have

. (Don't forget to change the bounds of integration by plugging them into for our equation for .)

### Example Question #23 : Definite Integrals

Evaluate .

**Possible Answers:**

Not possible without a calculator

**Correct answer:**

This integral isn't possible to integrate directly using antiderivatives, but we can still find its value by noticing that is an odd function , and that our limits of integration are negatives of each other.

Hence

. (Since is an odd function)

### Example Question #24 : Definite Integrals

Evaluate

**Possible Answers:**

None of the other answers.

**Correct answer:**

We can use u-substitution for this integral

. Start

Let , then , and our integral becomes

. (Don't forget to change the bounds of integration by plugging them into our equation for )

### Example Question #25 : Definite Integrals

Evaluate .

**Possible Answers:**

None of the other answers

**Correct answer:**

We proceed by using integration by parts.

. Start

Let , then we get

. Then using the integration by parts formula , we get

### Example Question #26 : Definite Integrals

Evaluate the following integral:

**Possible Answers:**

**Correct answer:**

To evaluate this integral, we must integrate by parts, according to the following formula:

So, we must assign our u and dv, and differentiate and integrate to find du and v, respectively:

The derivative and integral were found using the following rules:

,

Note that we ignore the constant of integration.

Now, use the above formula:

Note that both the product of u and v *and *the integral are being evaluated from zero to .

The integral was performed using the following rule:

Simplifying the above results, we get .

### Example Question #27 : Definite Integrals

Find the area between and between

**Possible Answers:**

**Correct answer:**

We can write this problem as:

Integrating:

By the fundamental theorem of calculus:

### Example Question #21 : Definite Integrals

**Possible Answers:**

**Correct answer:**

Compute the Indefinite Integral

Evaluate the integral

### Example Question #29 : Definite Integrals

Suppose , where is a constant

Find such that

**Possible Answers:**

**Correct answer:**

By the fundamental theorem of calculus:

### Example Question #30 : Definite Integrals

Evaluate this integral.

**Possible Answers:**

Answer not listed

**Correct answer:**

In order to evaluate this integral, first find the antiderivative of

In this case, .

The antiderivative is .

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

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