Integration of Exponential and Logarithmic Functions

Integration involving exponentials and logarithms is one of the most important topics in calculus because these functions appear in growth/decay, finance, physics, biology, and engineering.

This lesson covers:

1. Integrals of exponential functions

2. Integrals of logarithmic functions

3. Integrals involving bases other than e

4. Special cases and techniques

5. Practice examples

Integrals of Exponential Functions

Basic Rule

For the natural exponential function:

$$\int e^{u} dx = e^{u}+C$$

This is the simplest integral because the derivative of exponent is itself.

Exponential with coefficient

If the exponent contains a linear expression:

$$\int e^{ax} dx $$

Use u-substitution:

$$\int e^{ax} dx$$ let $$ax=u \longrightarrow a = \frac{du}{dx} \longrighrarrow dx = \frac{1}{a}du$$

then $$\int a^{ax} dx = \frac{1}{a} \int a^{u} du = \frac{1}{a} a^{u}+C = \frac{1}{a}a^{ax}+C$$

Example

$$\int e^{3x} dx$$ let $$ 3x = u \longrightarrow 3 dx = du \longrightarrow dx = \frac{1}{3}du$$

then $$\int e^{3x} dx = \frac{1}{3} \int e^{u} du = \frac{1}{3}e^{u}+C=\frac{1}{3}e^{3x}+C$$

Exponential with a constant base ($$a\neq e$$)

$$\int a^{x} dx = \frac{a^{x}}{\ln(a)}+C$$

This is because integration exponent with constant base derivate is :

$$\frac{d}{dx}(a^{x}) = a^{x}\ln(a) \longrightarrow \int a^{x} dx = \frac{a^{x}}{\ln(a)}+C$$

Example

$$\int 5^{x} dx = \frac{5^{x}}{\ln(5)}+C$$

Integrals of Logarithmic Functions

The basic integral everyone should know:

$$\int \ln(x) dx$$

Use integration by parts:

$$\int \ln(x) dx = x\ln(x)-x+C$$

integration by parts we will study next time.