# Solving the Integral of cos(2x)

Coming up next: Solving the Integral of cos(x)

### You're on a roll. Keep up the good work!

Replay
Your next lesson will play in 10 seconds
• 0:00 Steps to Solve
Save Save

Want to watch this again later?

Timeline
Autoplay
Autoplay
Speed

#### Recommended Lessons and Courses for You

Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we will find the integral of cos(2x) using integration by substitution in a step-by-step process. We will then go on to check our work using derivatives and the chain rule for derivatives.

## Steps to Solve

We are interested in finding the integral of cos(2x). To do this, we are going to make use of integration by substitution. Integration by substitution is an integration method that can be used on integrals that are in the form âˆ« f(g(x)) dx, and can be put in the following form:

âˆ« f(g(x)) â‹… g ' (x) dx

When we have an integral in this form, we can make a u-substitution where u = g(x), so du = g ' (x) dx. By making this substitution, we end up with the following:

âˆ« f(g(x)) â‹… g ' (x) dx = âˆ« f(u) du

Then, we can find the integral of f(u) and plug g(x) back in for u to get the integral. Hmmmâ€¦that sounds a little confusing. Let's see if writing this process out in steps makes it a bit more clear.

To find âˆ« f(g(x)) â‹… g ' (x) dx, we follow these steps.

1. Let u = g(x). Then du = g ' (x) dx, and dx = (1/g ' (x))du.
2. Plug these values into the integral appropriately to get it in the form âˆ« f(u) du or âˆ« a â‹… f(u) du, where a is a constant.
3. Evaluate the integral in terms of u.
4. Plug g(x) back in for u.

Alright, these steps don't seem so bad, but if you really want to learn this, it helps to practice it out in the form of problems, so let's find the integral of cos(2x) using this process!

There are a few more facts that we need to know to be able to find this integral, and those are as follows:

• The derivative of 2x is 2.
• The integral of cos(x) is sin(x) + C, where C is a constant.
• If a is a constant, then âˆ« a â‹… f(x) dx = aâˆ« f(x) dx.

Okay, let's get to work!

First, we notice that if we let f(x) = cos(x) and g(x) = 2x, then f(g(x)) = cos(2x), so we are finding the integral of f(g(x)). Recognizing this makes our substitution much easier, which brings us to our first step, and that is to let u = g(x) and du = g ' (x) dx.

Alright, we have that u = 2x and that dx = (1/2)du. The next step is to plug these values into our integral. That is, we plug in u for 2x, and we plug in (1/2)du for dx.

Moving on to the third step, we see that we want to find âˆ« (1/2)cos(u) du. To do this, we make use of our facts that we mentioned earlier and find:

We have that âˆ« (1/2)cos(u) du = (1/2)sin(u) + C, where C is a constant. We're almost there! Just one last step! We need to plug 2x back in for u to get (1/2)sin(2x) + C, where C is a constant.

#### Solution

The integral of cos(2x) is (1/2)sin(2x) + C, where C is a constant.

Well, we've got our answer, but there were a few steps involved, so we probably want to make sure we didn't make any mistakes along the way. In other words, it would be nice to be able to check that our answer is correct. Thankfully, we can use derivatives to do so!

To unlock this lesson you must be a Study.com Member.

### Register to view this lesson

Are you a student or a teacher?

#### See for yourself why 30 million people use Study.com

##### Become a Study.com member and start learning now.
Back
What teachers are saying about Study.com

### Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.