Exponential Function Integration

Integrating Exponential Functions

When you integrate an exponential function, look for the base and for any constant multiplying the exponent. The main idea is that exponentials stay exponential after integration, but the result may need a constant factor to adjust for the derivative of the exponent.

1) Identify the form

Common forms include:

• (\int e^x,dx)
• (\int a^x,dx)
• (\int e^{kx},dx)
• (\int a^{kx},dx)

2) Use the matching rule

• For (\int e^x,dx), the antiderivative is (e^x + C).
• For (\int e^{kx},dx), divide by (k): [ \int e^{kx},dx = \frac{1}{k}e^{kx}+C. ]
• For (\int a^x,dx), use the natural log of the base: [ \int a^x,dx = \frac{a^x}{\ln a}+C, \quad a>0,\ a\neq 1. ]
• For (\int a^{kx},dx), combine both adjustments: [ \int a^{kx},dx = \frac{a^{kx}}{k\ln a}+C. ]

3) Simplify carefully

Reduce coefficients and write the constant of integration (C) at the end.

4) Check your answer

Differentiate your result. You should recover the original integrand exactly. This is the best way to confirm the factor in front is correct.