How to Calculate Integrals of Exponential Functions

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  • 0:11 Quick Calculus Review
  • 0:27 First Example: e^x
  • 1:46 Second Example: cos(x) - e^x
  • 4:00 Lesson Summary
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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

The derivative of e^x is e^x
E to the X Explanation

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.

Solution for the first example problem
Exponential Function Integral Example 1

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