Integration of Exponential and Logarithmic Function

Exponential functions have the form f(x) = a^x or f(x) = ⅇ^x, where the variable is in the exponent. Logarithmic functions are the inverses of exponential functions, commonly written as log_b(x) or ln(x) (natural logarithm, base ⅇ). They are fundamental in modeling growth, decay, and many natural phenomena.

The derivatives are:

• For the natural exponential function: ⅆ / ⅆx (ⅇ^x) = ⅇ^x
• For a general exponential function (a > 0, a ≠ 1): ⅆ / ⅆx (a^x) = a^x · ln(a)

These simple formulas make exponential functions particularly important in calculus.

Based on the derivative rules, the integral formulas are:

• For the natural exponential function: ∫ ⅇ^x ⅆx = ⅇ^x + C
• For a general exponential function (a > 0, a ≠ 1): ∫ a^x ⅆx = a^xln(a) + C

Remember to include the constant of integration, C, for indefinite integrals.

The derivatives are:

• For the natural logarithm: ⅆ / ⅆx (ln(x)) = 1x (for x > 0)
• For a general logarithm (b > 0, b ≠ 1): ⅆ / ⅆx (log_b(x)) = 1x · ln(b) (for x > 0)

The general logarithm derivative can be found using the change of base formula (log_b(x) = ln(x) / ln(b)).

While the derivative of ln(x) is simple, the integral of ln(x) itself is not immediately obvious from derivative rules and requires integration by parts:

• For the natural logarithm: ∫ ln(x) ⅆx = x ln(x) − x + C (for x > 0)

The integral of log_b(x) can be found by first using the change of base formula (log_b(x) = ln(x) / ln(b)) and then integrating ln(x).