Antiderivatives by substitution of variables - AP Calculus AB
Card 1 of 649
Use a change of variable (aka a u-substitution) to evaluate the integral,
Use a change of variable (aka a u-substitution) to evaluate the integral,
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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,
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,
Now differentiate with respect to
Looking at equation (2), we can solve for
Now proceed with the integration with respect to
Now write the result in terms of
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Solve the following integral using substitution:
Solve the following integral using substitution:
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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.
3. 
4. 
5. 
6. 
7. 
8. 
9. 
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
The
The specific steps are as follows:
1.
2.
3.
4.
5.
6.
7.
8.
9.
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Solve the following integral using substitution:
Solve the following integral using substitution:
Tap to reveal answer
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. 
To solve the integral, we have to simplify it by using a variable
1.
2.
3.
4.
5.
6.
7.
8.
← Didn't Know|Knew It →
Solve the following integral using substitution:
Solve the following integral using substitution:
Tap to reveal answer
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. 
To solve the integral, we have to simplify it by using a variable
1.
2.
3.
4.
5.
6.
7.
← Didn't Know|Knew It →
Solve the following integral using substitution:
Solve the following integral using substitution:
Tap to reveal answer
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. 
To solve the integral, we have to simplify it by using a variable
1.
2.
3.
4.
5.
6.
7.
← Didn't Know|Knew It →
Solve the following integral using substitution:
Solve the following integral using substitution:
Tap to reveal answer
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. 
To solve the integral, we have to simplify it by using a variable
1.
2.
3.
4.
5.
6.
7.
← Didn't Know|Knew It →
Solve the following integral using substitution:
Solve the following integral using substitution:
Tap to reveal answer
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. 
To solve the integral, we have to simplify it by using a variable
1.
2.
3.
4.
5.
6.
7.
← Didn't Know|Knew It →
Solve the following integral using substitution:
Solve the following integral using substitution:
Tap to reveal answer
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. 
To solve the integral, we have to simplify it by using a variable
1.
2.
3.
4.
5.
6.
7.
8.
← Didn't Know|Knew It →
Solve the following integral using substitution:
Solve the following integral using substitution:
Tap to reveal answer
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. 
8. 
To solve the integral, we have to simplify it by using a variable
1.
2.
3.
4.
5.
6.
7.
8.
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Solve:
Solve:
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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
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
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Use u-subtitution to fine
Use u-subtitution to fine
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Let 
Then 
Now we can subtitute
Now we substitute back
Let
Then
Now we can subtitute
Now we substitute back
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Evaluate 
Evaluate
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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
.
We can use substitution for this integral.
Let
then
Multiplying this last equation by
Now we can make our substitutions
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This is a u-substitution integral. We need to substitute the new function, which is modifying our base function (the exponential).
, but instead of that, our problem is 
. We can solve this integral by completing the substitution.
Now, we can replace everything in our integrand and rewrite in terms of our new variables:
.
Remember to plug your variable back in and include the integration constant since we have an indefinite integral.
This is a u-substitution integral. We need to substitute the new function, which is modifying our base function (the exponential).
Now, we can replace everything in our integrand and rewrite in terms of our new variables:
Remember to plug your variable back in and include the integration constant since we have an indefinite integral.
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This is a u-substitution integral. We need to choose the following substitutions:
Now, we can replace our original problem with our new variables:
In the last step, we need to plug in our original function and add the integration constant.
This is a u-substitution integral. We need to choose the following substitutions:
Now, we can replace our original problem with our new variables:
In the last step, we need to plug in our original function and add the integration constant.
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This is a hidden u-substitution problem! Because we have a function under our square root, we cannot just simply integrate it. Therefore, we need to choose the function under the square root as our substitution variable!
Now, let us rewrite our original equation in terms of our new variable!
.
This is a hidden u-substitution problem! Because we have a function under our square root, we cannot just simply integrate it. Therefore, we need to choose the function under the square root as our substitution variable!
Now, let us rewrite our original equation in terms of our new variable!
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This is a hidden u-substitution problem! Remember, to use substitution, we need to have an integral where a function and its derivative live inside. If you look closely, you will see we have just that!
Now, rewrite the integral, and integrate:
This is a hidden u-substitution problem! Remember, to use substitution, we need to have an integral where a function and its derivative live inside. If you look closely, you will see we have just that!
Now, rewrite the integral, and integrate:
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This is a u-substitution problem. We need to find a function and its derivative in the integral.
Now, replace your variables, and integrate.
.
This is a u-substitution problem. We need to find a function and its derivative in the integral.
Now, replace your variables, and integrate.
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This problem is an application of the u-substitution method.
Now, be careful that you replace everything in the original integral in terms of our new variables. This includes the 
term!
.
This problem is an application of the u-substitution method.
Now, be careful that you replace everything in the original integral in terms of our new variables. This includes the
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To simplify the integral, we need to substitute new variables:
Now, we can replace our original variables, and integrate!
.
To simplify the integral, we need to substitute new variables:
Now, we can replace our original variables, and integrate!
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Integrate:
Integrate:
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To integrate, we must make the following substitution:
The derivative was found using the following rule:
Now, we rewrite the integral in terms of u and solve:
The integral was found using the following rule:
Finally, replace u with our original x term:
To integrate, we must make the following substitution:
The derivative was found using the following rule:
Now, we rewrite the integral in terms of u and solve:
The integral was found using the following rule:
Finally, replace u with our original x term:
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