Question

write integration formulae for exponential functions

Answer 1

The integration formula for exponential functions can be expressed as follows:

1. **Basic Exponential Function:**
\[
\int e^{x} \, dx = e^{x} + C
\]
where \( C \) is the constant of integration.

2. **Exponential Function with a Constant Factor:**
If you have an exponential function with a constant factor in the exponent, say \( ax \), the integral is:
\[
\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C
\]
where \( a \neq 0 \).

3. **Integration of Exponential Functions with Base \( b \):**
If you are integrating an exponential function with a base other than \( e \) (where \( b > 0, b \neq 1 \)):
\[
\int b^{x} \, dx = \frac{b^{x}}{\ln(b)} + C
\]

4. **Exponential Functions with Additional Terms:**
For more complex functions, such as integrals of the form \( e^{g(x)} \cdot g'(x) \), the formula applies directly:
\[
\int e^{g(x)} \cdot g'(x) \, dx = e^{g(x)} + C
\]

These formulae can be used to integrate various forms of exponential functions encountered in calculus.