# How to Calculate Integrals of Exponential Functions

Coming up next: How to Solve Integrals Using Substitution

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

Replay
Your next lesson will play in 10 seconds
• 0:11 Quick Calculus Review
• 0:27 First Example: e^x
• 1:46 Second Example: cos(x) - e^x
• 4:00 Lesson Summary
Save Save

Want to watch this again later?

Timeline
Autoplay
Autoplay
Speed

#### Recommended Lessons and Courses for You

Lesson Transcript
Instructor: Kelly Sjol
Exponential functions are so predictable. It doesn't matter how many times you differentiate e^x, it always stays the same. In this lesson, learn what this means for finding the integrals of such boring functions!

## Quick Calculus Review

Let's review. If you want to find a definite integral of f(x) from x=a to x=b, you need to find the anti-derivative of f(x) evaluated at b and a and take the difference, F(b) - F(a).

## First Example: e^x

What if your function is f(x)=e^x. What is the integral of e^x dx? Remember that e^x is the exponential, some number e (roughly 2.7), to the x power. If you take the derivative of e^x, you get back e^x. It's one of those functions that, no matter how many times you take the derivative, you still get back e^x. So it makes a lot of sense that the integral of e^x dx is nothing more than e^x + some constant, C. If you take the derivative of e^x + C, you end up with e^x, because the derivative of C becomes zero.

So let's actually do an example. Let's calculate the derivative of e^x from x= -1 to x=1. We're going to use the fundamental theorem of calculus, which says that I need to know the anti-derivative of e^x and evaluate it from -1 to 1. That anti-derivative is just e^x, because the derivative of e^x is e^x. So e^x evaluated from -1 to 1 is e^1 - e^(-1). That's the same as e - 1/e, because e^(-1) is the same as 1/e.

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.