The integration formula for exponential functions can be expressed as follows:
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Basic Exponential Function: \[ \int e^{x} , dx = e^{x} + C \] where \( C \) is the constant of integration.
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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 \( a \neq 0 \).
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Integration of Exponential Functions with Base \( b \): If you are integrating an exponential function with a base other than \( e \) (where \( b > 0, b \neq 1 \)): \[ \int b^{x} , dx = \frac{b^{x}}{\ln(b)} + C \]
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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.