# AP Calculus AB : Antiderivatives by substitution of variables

## Example Questions

### Example Question #11 : Antiderivatives By Substitution Of Variables

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:

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### Example Question #12 : Antiderivatives By Substitution Of Variables

Explanation:

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:

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Remember to plug your variable back in and include the integration constant since we have an indefinite integral.

### Example Question #13 : Antiderivatives By Substitution Of Variables

Explanation:

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.

### Example Question #14 : Antiderivatives By Substitution Of Variables

Explanation:

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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### Example Question #15 : Antiderivatives By Substitution Of Variables

Explanation:

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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### Example Question #16 : Antiderivatives By Substitution Of Variables

Explanation:

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!

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### Example Question #17 : Antiderivatives By Substitution Of Variables

Explanation:

To simplify the integral, we need to substitute new variables:

Now, we can replace our original variables, and integrate!

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### Example Question #18 : Antiderivatives By Substitution Of Variables

Explanation:

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:

### Example Question #19 : Antiderivatives By Substitution Of Variables

Integrate:

Explanation:

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:

Integrate: