# AP Calculus AB : Antiderivatives by substitution of variables

## Example Questions

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### Example Question #1 : Antiderivatives By Substitution Of Variables

Use a change of variable (aka a u-substitution) to evaluate the integral,

Explanation:

Integrals such as this are seen very commonly in introductory calculus courses. It is often useful to look for patterns such as the fact that the polynomial under the radical in our example, , happens to be one order higher than the factor outside the radical,  You know that if you take a derivative of a second order polynomial you will get a first order polynomial, so let's define the variable:

(1)

Now differentiate with respect to  to write the differential for

(2)

Looking at equation (2), we can solve for , to obtain  . Now if we look at the original integral we can rewrite in terms of

Now proceed with the integration with respect to

Now write the result in terms of  using equation (1), we conclude,

### Example Question #2 : Antiderivatives By Substitution Of Variables

Use u-subtitution to fine

Explanation:

Let

Then

Now we can subtitute

Now we substitute back

### Example Question #3 : Antiderivatives By Substitution Of Variables

Evaluate

Explanation:

We can use substitution for this integral.

Let ,

then .

Multiplying this last equation by , we get .

Now we can make our substitutions

. Start

. Swap out  with , and  with . Make sure you also plug the bounds on the integral into  for  to get the new bounds.

. Factor out the .

. Integrate (absolute value signs are not needed since .)

. Evaluate

.

### Example Question #51 : Techniques Of Antidifferentiation

Solve the following integral using substitution:

Explanation:

To solve the integral, we have to simplify it by using a variable u to substitute for a variable of x.

For this problem, we will let u replace the expression .

Next, we must take the derivative of u. Its derivative is .

Next, solve this equation for dx so that we may replace it in the integral.

Plug  in place of and  in place of  into the original integral and simplify.

The  in the denominator cancels out the remaining  in the integral, leaving behind a . We can pull the  out front of the integral. Next, take the anti-derivative of the integrand and replace u with the original expression, adding the constant  to the answer.

The specific steps are as follows:

1.

2.

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### Example Question #52 : Techniques Of Antidifferentiation

Solve the following integral using substitution:

Explanation:

To solve the integral, we have to simplify it by using a variable  to substitute for a variable of . For this problem, we will let u replace the expression . Next, we must take the derivative of u. Its derivative is . Next, solve this equation for  so that we may replace it in the integral. Plug  in place of  and  in place of  into the original integral and simplify. The  in the denominator cancels out the remaining  in the integral, leaving behind a . We can pull the  out front of the integral. Next, take the anti-derivative of the integrand and replace  with the original expression, adding the constant  to the answer. The specific steps are as follows:

1.

2. =

3.

4.

5.

6.

7.

8.

### Example Question #53 : Techniques Of Antidifferentiation

Solve the following integral using substitution:

Explanation:

To solve the integral, we have to simplify it by using a variable  to substitute for a variable of . For this problem, we will let u replace the expression . Next, we must take the derivative of . Its derivative is .  Next, solve this equation for  so that we may replace it in the integral. Plug  in place of  and  in place of  into the original integral and simplify. The  in the denominator cancels out the remaining  in the integral, leaving behind a . Next, take the anti-derivative of the integrand and replace  with the original expression, adding the constant  to the answer. The specific steps are as follows:

1.

2.

3.

4.

5.

6.

7.

### Example Question #54 : Techniques Of Antidifferentiation

Solve the following integral using substitution:

Explanation:

To solve the integral, we have to simplify it by using a variable  to substitute for a variable of . For this problem, we will let u replace the expression . Next, we must take the derivative of . Its derivative is . Next, solve this equation for  so that we may replace it in the integral. Plug  in place of  and  in place of  into the original integral and simplify. The  in the denominator cancels out the remaining  in the integral, leaving behind a .  We can pull the  out front of the integral. Next, take the anti-derivative of the integrand and replace  with the original expression, adding the constant  to the answer. The specific steps are as follows:

1.

2.

3.

4.

5.

6.

7.

### Example Question #55 : Techniques Of Antidifferentiation

Solve the following integral using substitution:

Explanation:

To solve the integral, we have to simplify it by using a variable  to substitute for a variable of . For this problem, we will let u replace the expression . Next, we must take the derivative of . Its derivative is . Next, solve this equation for  so that we may replace it in the integral. Plug  in place of  and  in place of  into the original integral and simplify. The  in the denominator cancels out the remaining  in the integral, leaving behind a . We can pull the  out front of the integral. Next, take the anti-derivative of the integrand and replace  with the original expression, adding the constant  to the answer. The specific steps are as follows:

1.

2.

3.

4.

5.

6.

7.

### Example Question #56 : Techniques Of Antidifferentiation

Solve the following integral using substitution:

Explanation:

To solve the integral, we have to simplify it by using a variable  to substitute for a variable of . For this problem, we will let u replace the expression . Next, we must take the derivative of . Its derivative is . Next, solve this equation for  so that we may replace it in the integral. Plug  in place of  and  in place of  into the original integral and simplify. The  in the denominator cancels out the remaining  in the integral. Next, take the anti-derivative of the integrand and replace  with the original expression, adding the constant  to the answer. The specific steps are as follows:

1.

2.

3.

4.

5.

6.

7.

### Example Question #57 : Techniques Of Antidifferentiation

Solve the following integral using substitution:

Explanation:

To solve the integral, we have to simplify it by using a variable  to substitute for a variable of . For this problem, we will let u replace the expression . Next, we must take the derivative of . Its derivative is . Next, solve this equation for  so that we may replace it in the integral. Plug  in place of  and  in place of  into the original integral and simplify. The  in the denominator cancels out the remaining  in the integral. Next, take the anti-derivative of the integrand and replace  with the original expression, adding the constant  to the answer. The specific steps are as follows:

1.

2.

3.

4.

5.

6.

7.

8.

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