# AP Calculus AB : Antiderivatives by substitution of variables

## Example Questions

### Example Question #221 : Integrals

Integrate:

Explanation:

To integrate, the following substitution was made:

Now, we rewrite the integral in terms of u and integrate:

The following rule was used for integration:

Finally, rewrite the final answer in terms of our original x term:

### Example Question #21 : Antiderivatives By Substitution Of Variables

Evaluate the integral

Explanation:

We can make a u substitution in the following way:

, and therefore

Simplifying the integral, we get

Rewriting in terms of x, we get

### Example Question #22 : Antiderivatives By Substitution Of Variables

Solve:

Explanation:

To integrate, we must make the following substitution:

Rewriting the integral in terms of u and integrating, we get

The following rule was used for integration:

Replacing u with our original x term, we get

### Example Question #23 : Antiderivatives By Substitution Of Variables

Solve:

Explanation:

To integrate, we must make the following substitution:

Rewriting the integral in terms of u and integrating, we get

The following rule was used for integration:

Replacing u with our original x term, we get

### Example Question #24 : Antiderivatives By Substitution Of Variables

Calculate:

Explanation:

Rewrite the integrand as follows:

.

The integral can be rewritten as

Now, use -substitution, setting . It follows that

The limits of integration can be rewritten as

The integral becomes

Integrate:

,

the correct response.

### Example Question #25 : Antiderivatives By Substitution Of Variables

Evaluate the following integral

Explanation:

The integral can be solved by using a variable substitution

Replacing  with , we get our final answer, which is

### Example Question #26 : Antiderivatives By Substitution Of Variables

Evaluate the following integral

Explanation:

We solve the problem by making a variable substitution

The integral then becomes

Substituting  for , we get our final answer

### Example Question #71 : Techniques Of Antidifferentiation

Solve:

Explanation:

To integrate, it is easiest to break the integral into the sum of two integrals:

To integrate the first integral, we must make the following substitution:

The derivative was found using the rule itself.

Rewriting the first integral in terms of u and integrating, we get

which was found using the rule itself.

Replacing u with our original x term, we get

The second integral is equal to

and was found using the following rule:

Adding our two results, and combining the two constants of integration into a single integration constant, we get

### Example Question #231 : Integrals

Calculate the integral in the following expression:

Explanation:

The simplest path to follow when trying to integrate a lot of trig expressions is often to put everything in terms of sin(x) and cos(x). Doing this for the above expressions yields:

Next, we look for expressions that we know how to integrate, based on the following facts:

And, of course, the simpler derivatives of sin(x) and cos(x).

Looking for these above expressions in our integral, we note that

Breaking this up into two integrals, we see that the second immediately can be simplified into -csc(x) + C. The first, while a bit more tricky, just requires you to realize that sec(x)tan(x) is the derivative of sec(x). Thus, if you use a substitution of variables (u-sub) with u = sec(x), you will get,

and

In this form, it is clear to see that the integral is just   where u = sec(x)

Combining our two integrals, we get a final answer of

### Example Question #29 : Antiderivatives By Substitution Of Variables

Evaluate the integral