Integrating Exponential Functions
Learn integrating exponential functions for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-05). Because \(e^x\) is unchanged by differentiation, it is also its own primitive, and that makes exponential integrals some of the simplest in the course.
You will learn to integrate \(e^x\) and \(e^{ax + b}\) by dividing by the coefficient of \(x\), handle other bases using \(\ln a\), recognise forms with the derivative of the power out the front, and evaluate definite integrals — needed for growth, decay and area problems.
Theory
\(e^{x}\) integrates back to \(e^{x}\); \(e^{ax}\) picks up a \(\dfrac1a\). This Year 12 Mathematics Advanced topic (MAV-12-05) integrates exponential functions.
\(\displaystyle\int e^{x}dx=e^{x}+C\) and \(\displaystyle\int e^{ax}dx=\dfrac1a e^{ax}+C\): divide by the coefficient of \(x\) in the exponent. Constant multiples stay out the front.
Method
- Keep the exponential.
- Divide by the coefficient of \(x\) in the exponent.
- Add \(+C\) (or evaluate limits for a definite integral).
| \(\ \) | \(=\) | \(\dfrac12 e^{2x}+C\) |
| \(\ \) | \(=\) | \(e^{x}+C\) |
| \(\ \) | \(=\) | \(-3e^{-x}+C\) |
| \(\ \) | \(=\) | \(\big[e^{x}\big]_0^1=e-1\) |
Common pitfalls
Frequently asked questions
What is the integral of e to the x?
It is e to the x plus C, because e to the x is its own antiderivative.
How do you integrate e to the 2x?
Divide by the coefficient 2: the integral is one half e to the 2x plus C.
What is the integral of e to the minus x?
It is minus e to the minus x plus C, because you divide by minus 1.
What is the area under e to the x from 0 to 1?
It is e to the 1 minus e to the 0, which equals e minus 1.