Integrals of exponential functions

$e^x$ stays $e^x$ whether you differentiate or integrate. For base $a$, differentiation multiplies by $\ln a$, so integration divides by it.

When the exponent is linear, as in $e^{2x}$, remember to divide by the coefficient of $x$.

Examples

1. 01What is the value?$\displaystyle \int_{0}^{1} e^{x}\,dx$AnswerHide

   Answer$\displaystyle e-1$
2. 02Integrate$\displaystyle \int e^x \, dx$AnswerHide

   Answer$\displaystyle e^x + C$
3. 03What goes in □?$\displaystyle \int a^{x}\,dx = \frac{a^{x}}{\square}$AnswerHide

   Answer$\displaystyle \log_e a$
4. 04What is this?$\displaystyle \int e^{x}\,dx$AnswerHide

   Answer$\displaystyle e^{x}+C$
5. 05What is this?$\displaystyle \int 3^{x}\,dx$AnswerHide

   Answer$\displaystyle \frac{3^{x}}{\log_e 3}+C$
6. 06What is this?$\displaystyle \int 5^{x}\,dx$AnswerHide

   Answer$\displaystyle \frac{5^{x}}{\log_e 5}+C$

FAQ

Q1What is $\int e^{2x}\,dx$?Show answerHide

A

$\frac{1}{2}e^{2x}+C$. Differentiation multiplies by the inside coefficient 2, so integration divides by 2.

Practice this pattern in the app, 5 problems a day.

Published 2026-09-03 · Updated 2026-09-05