### All Calculus 2 Resources

## Example Questions

### Example Question #72 : Solving Integrals By Substitution

Evaluate the integral:

**Possible Answers:**

**Correct answer:**

A u-substitution would properly simplify the integrand, where

Now, the problem can be rewritten entirely in terms of u:

The problem may seem finished, but the original integrand was expressed in terms of . Therefore, the final answer is, in fact:

### Example Question #77 : Solving Integrals By Substitution

Evaluate the integral:

**Possible Answers:**

**Correct answer:**

There are no apparent substitutions for solving this integral, but the integral can be expressed as the sum of two separate integrals because this is a property of indefinite integrals.

The first integral can be solved with a simple u-substitution where .

The integral can be rewritten as:

To finally solve this, there is no other way to do so other than knowing the following:

Finally, the answer must be expressed in terms of :

The second integral is a bit more complicated. It can be noted that the second integral resembles the following:

Specifically, . The second integral can be rewritten as:

With each separate integral found, the answers can be added to equal the original integral:

### Example Question #661 : Finding Integrals

Evaluate the integral:

**Possible Answers:**

**Correct answer:**

To solve this, you can use a u-substitution where

.

Now, the integrand can be completely rewritten in terms of u:

Before trying to further solve this integral in a more complicated way, by remembering what u equals, the integrand can be rewritten as:

The integral was taken by using the following formula:

The original problem was in terms of x. Therefore, the final answer is:

### Example Question #80 : Solving Integrals By Substitution

Use trigonometric substitution to set up the given integral in terms of . Do not evaluate the integral:

**Possible Answers:**

**Correct answer:**

The trig sub is used to redefine the integral in terms of a different variable, , to make its evaluation possible.

For the given integral, following steps are important to redefine it using trigonometric substitution:

1) Find out, which trigonometric identity is best for given integral. In our case, the integrand is: . Therefore, is an appropriate identity.

2) Redefine the bounds of integration:

3) Change to . Differentiating from part (1):

4) Rewrite the integral in terms of :

### Example Question #81 : Solving Integrals By Substitution

Calculate the following integral:

**Possible Answers:**

**Correct answer:**

Solve via integration by parts. Make the following substitutions:

.

Plug in substitutions: .

Solve via integration by parts again. Make the following substitutions:

.

Plug in substitutions: .

Simplify: .

Plug this integral back into our original equation:

Simplify: .

Add to both sides of the equation: .

Factor out of the right side of the equation:

Divide both sides of the equation by :

### Example Question #81 : Solving Integrals By Substitution

Calculate the following integral:

**Possible Answers:**

**Correct answer:**

Factor out a sinx:

Apply Pythagorean identity to : .

Making the following substitutions:

Apply substitutions:

Solve integral:

Convert u back to x:

### Example Question #83 : Solving Integrals By Substitution

Calculate the following integral:

**Possible Answers:**

**Correct answer:**

Make the following substitution: .

Plug the substitution into the integrand: .

Factor out from the denominator, and simplify: .

Apply Pythagorean identity to the denominator, and simplify:.

Solve integral:.

Use original substitution to solve for : .

Plug value for back into solution for integral: .

### Example Question #82 : Solving Integrals By Substitution

Evaluate the following indefinite integral:

**Possible Answers:**

**Correct answer:**

This integral, , is a classic integration by substation problem. This is indicated by the presence of a composite function in which both a function and its derivative are present.

For this we need to let our variable , substituting this into our integral we produce the following:

and by substituting back in for u we find our final answer to be:

.

### Example Question #85 : Solving Integrals By Substitution

Evaluate the integral and use the sine function to make an appropriate trigonometric substitution.

**Possible Answers:**

**Correct answer:**

** (1)**

** **

To compute this integral we will use a trigonometric substitution. First we need to do some algebra to write the integrated in a more suitable form,

**(2)**

Now we can apply a trigonometric substitution.

(3)

(4)

Notice we've chosen this substitution so that the radicand in equation (2) will conveniently reduce as follows,

So now we have , we need to convert back to , use equation (3),

### Example Question #671 : Finding Integrals

Integrate

**Possible Answers:**

**Correct answer:**

**(1)**

**1)** **Simplify with a substitution. **

It is often necessary to define a new variable , carefully chosen so that rewriting the integrand in terms of this new variable will make integration easier. In this case, the obvious variable to introduce will be defined by,

** (2)**

** (3)**

Use equations** (2)** and** (3)** to rewrite **(1).**

**2) Use integration by parts**

To compute use integration by parts. Ignore the constant out front for the moment,

** (4)**

Define and and insert into the equation **(4). **

,

** (5)**

Let's factor the non-constant terms in equation** (5),** this will make the result easier to express when we convert back to

We previously had a constant in front of the integral,

Now we can write the integral terms of the original variable by substituting equation** (2)** into the previous expression to obtain,