# AP Calculus AB : Antiderivatives by substitution of variables

## Example Questions

### Example Question #31 : Antiderivatives By Substitution Of Variables

Many integrals are unsolvable, the techniques you learn are for special cases of integrals, and are used in the scenarios where the integral is possible to do by hand.

Evaluate the integral by using integration by substitution.

Explanation:

So, unlike with derivatives, there is no equivalent for the chain rule of the product rule for integrals. Anytime you see a function within a function (a compound function) and/or the product of more than one function inside an integral, you can't just take the integral. In this case, we are able to perform a substitution. So the goal with a substitution is to replace all of the x variables (including dx) with another variable (we use u, and du).

So we can set

which gives us

So this changes our integral to:

Which just turns out to:

the C is added because we have no bounds for the integral.

After replacing u for what is originally was, we end up with:

### Example Question #32 : Antiderivatives By Substitution Of Variables

Many integrals are unsolvable, the techniques you learn are for special cases of integrals, and are used in the scenarios where the integral is possible to do by hand.

Evaluate this integral using substitution.

Explanation:

So, unlike with derivatives, there is no equivalent for the chain rule of the product rule for integrals. Anytime you see a function within a function (a compound function) and/or the product of more than one function inside an integral, you can't just take the integral. In this case, we are able to perform a substitution. So the goal with a substitution is to replace all of the x variables (including dx) with another variable (we use u, and du).

For this problem, we set

This gives us

So the thing to note with substitutions is that the  that you choose has to have a derivative that is also in the integral. If the  had not been in the integral, then this substitution would not work.

So making these substitutions gives:

solving this integral gives:

Substituting the u back into the equation gives us a final answer of:

### Example Question #33 : Antiderivatives By Substitution Of Variables

Explanation:

We are going to use U-substitution

Looking at the original

Let

Then

And now we can use our anti derivative rules (don't forget your constant!)

Finally, substitute back in for u

### Example Question #34 : Antiderivatives By Substitution Of Variables

Integrate:

Explanation:

To evaluate the integral, we must make the following substitution:

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

The integral was found using the following rule:

To finish, rewrite the answer in terms of x:

### Example Question #35 : Antiderivatives By Substitution Of Variables

Solve:

Explanation:

To integrate, the following substitution is made:

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

The integral was performed using the following rule:

Finally, replace u with the original term (containing x):

### Example Question #36 : Antiderivatives By Substitution Of Variables

Solve:

Explanation:

To integrate, we must perform the following substitution:

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

The integral was performed using the identical rule (the constant in front doesn't change the integral).

Finally, rewrite the result in terms of x:

### Example Question #37 : Antiderivatives By Substitution Of Variables

Integrate:

Explanation:

To integrate, we can split the integral into two integrals:

The first integral is equal to

and was found using the following rule:

The second integral can be made easier with the following substitution:

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

The integral was performed using the identical rule.

Next, we rewrite our answer in terms of x, and add it to the first integral's result, combining the two integration constants into a single one:

### Example Question #38 : Antiderivatives By Substitution Of Variables

Integrate:

Explanation:

To integrate, we can split the integral up (the property of linearity allows us to do this):

The first integral is equal to

and was found using the following rule:

The second integral can be solved after the following substitution is made:

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

The integral was solved using the identical rule.

Next, rewrite the answer in terms of x:

Finally, add this to the first result to get our final answer:

Note that all of the integration constants were combined to make a single constant.

### Example Question #39 : Antiderivatives By Substitution Of Variables

Integrate:

Explanation:

To integrate, the following substitution must be made:

The following rule was used to find the derivative:

Next, rewrite the integral in terms of u and integrate:

The integral was performed using the identical rule.

Finally, rewrite the integral in terms of x by replacing u:

Integrate: