Solving the Integral of ln(x)

Solving the Integral of ln(x)
Coming up next: Finding the Antiderivative of 1/cos(x)

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

Take Quiz Watch Next Lesson
 Replay
Your next lesson will play in 10 seconds
  • 0:04 Steps to Solve
  • 1:25 Application
  • 2:21 Solution
  • 2:28 Integration by Parts
Add to Add to Add to

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Log in or Sign up

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.

Finding the integral of ln(x) requires the process of integration by parts. This lesson will show how to find the integral of ln(x) using integration by parts and explain how and why integration by parts works.

Steps to Solve

The integral of ln(x) may look simple, but it's actually a bit involved. To find this integral, we have to use integration by parts. This process is used to find the integral of a product of functions. The formula we use for integration by parts is as follows:


intlnx1


Now you may look at our problem, solve the integral of ln(x), and wonder how this is a product of functions. Well, we can think of the integral of ln(x) as the integral of ln(x)*1. This way, we have a product of the functions f(x) = ln(x) and g(x) = 1.


intlnx2


Okay, now that we've cleared that up, let's look at the steps involved when using integration by parts.

  1. Identify your two functions u and dv. This is usually the hardest step. One thing to help with this step is to keep in mind that we want u to be a function that is easy to find the derivative of, and we want dv to be a function for which it's easy to find the integral. Put the dx from the original integral with whichever function you named as dv.
  2. Find du, or the derivative of u, and find v, or the integral of dv.
  3. Plug u, v, du, and dv into the integration according to the parts formula and simplify.

Application

Let's apply these steps to the integral of ln(x). We know our two functions are ln(x) and 1. Since the derivative of ln(x) is well-known as 1/x, it would probably be a good idea to let u = ln(x). Similarly, the integral of 1 is well-known as x + C, where C is a constant. Thus, we will let dv = 1dx. It's important to note that we don't include the constants when finding different integrals during the solving process. This is because the constants that would show up throughout will all be taken care of at the end of the process when we have our final constant.

At this point, we've actually completed steps 1 and 2, and we have our u, du, v, and dv.

u = ln(x) dv = 1dx
du = (1/x)dx v = x

All we have to do now is plug the results into our formula and simplify.


intlnx3


Solution

We see that the integral of ln(x) is xln(x) - x + C.


intlnx4


Integration by Parts

So we've found the integral of ln(x), but the use of integration by parts may be new to you, and it may have left you with some questions. To remedy this, let's take a closer look at integration by parts.

As we said, integration by parts is used to find the integral of products of functions. We can actually derive the formula for integration by parts from the product rule for derivatives. Let's see how this is done.

We'll start with the product formula for derivatives.


intlnx5


Next, we'll integrate both sides of the function. You may wonder why, but this approach will all become more and more clear as we move along.


inlnx6


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

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back
What teachers are saying about Study.com
Try it risk-free for 30 days

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.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create An Account
Support