Integral of Exponential
Antiderivative of the exponential function.
Formula
Variables
Example
e^x integrates to itself
How it works
The exponential function e^x is its own derivative, so an antiderivative must also be e^x, just with an added constant C to represent all possible vertical shifts. In other words, if you differentiate e^x + C, you get back e^x. This makes the integral especially useful because it reverses the simplest growth process in calculus.
When to use it
Use this when you need an antiderivative of e^x in calculus problems, especially in integration, solving differential equations, or modeling exponential growth and decay. It also appears whenever a more complicated expression contains e^x and you want to integrate by recognizing that structure.
Worked example, step by step
Find \(\int 7e^x\,dx\). Step 1: Pull out the constant 7. \(\int 7e^x\,dx = 7\int e^x\,dx\) Step 2: Use the basic exponential integral rule. \(7\int e^x\,dx = 7(e^x + C)\) Step 3: Write the result with a single constant. \(7e^x + C\) Final answer: \(\int 7e^x\,dx = 7e^x + C\)
Common mistakes to avoid
- Forgetting the +C, which means the antiderivative is incomplete
- Writing the result as x e^x instead of e^x
- Treating the integral of e^x as a power rule problem, which it is not
Did You Know?
Integrating e^x simply returns e^x, the calculus counterpart of its self-differentiating magic.
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