# 5.6: Integrals Involving Exponential and Logarithmic Functions

- Page ID
- 43663

Learning Objectives

- Integrate functions involving exponential functions.
- Integrate functions involving logarithmic functions.

Exponential and logarithmic functions are used to model population growth, cell growth, and financial growth, as well as depreciation, radioactive decay, and resource consumption, to name only a few applications. In this section, we explore integration involving exponential and logarithmic functions.

## Integrals of Exponential Functions

## Integrals Resulting in Natural Logarithms

Integrating functions of the form \(f(x)=\frac{1}{x}\) result in the absolute value of the natural log function, as shown in the following rule.

Rule: Integration Formulas Resulting in Natural Logarithms

The following formulas can be used to evaluate integrals involving logarithmic functions.

\[\begin{align*} ∫\frac{1}{x}\,dx &= \ln |x|+C \\[4pt] ∫\frac{u'(x)}{u(x)}\,dx &= \ln |u(x)|+C & & \text{which is another way to write,} \\[4pt] ∫\frac{1}{u}\,du &= \ln |u|+C \end{align*}\]

Example \(\PageIndex{9}\): Finding an Antiderivative Involving \(\ln x\)

Find the antiderivative of the function \(\dfrac{3}{x−10}. \)

**Solution**

First factor the \(3\) outside the integral symbol. Then use the \(\dfrac{u'}{u}\) rule. Thus,

\[∫\dfrac{3}{x−10}\,dx=3∫\dfrac{1}{x−10}\,dx=3∫\dfrac{1}{u}\,du=3\ln |u|+C=3\ln |x−10|+C,\quad x≠10. \nonumber\]

See Figure \(\PageIndex{3}\).

Exercise \(\PageIndex{8}\)

Find the antiderivative of \(\dfrac{1}{x+2}.\)

**Hint**-
Follow the pattern from Example \(\PageIndex{9}\) to solve the problem.

**Answer**-
\(\displaystyle \int \dfrac{1}{x+2}\,dx = \ln |x+2|+C\)

Example \(\PageIndex{10}\): Finding an Antiderivative of a Rational Function

Find the antiderivative of \(\dfrac{2x^3+3x}{x^4+3x^2}. \)

**Solution**

This can be rewritten as \(\displaystyle ∫\dfrac{2x^3+3x}{x^4+3x^2}\,dx.\) Use substitution.

Let \(u=x^4+3x^2\), then \(du=(4x^3+6x)\,dx.\) Alter \(du\) by factoring out the \(2\). Thus,

\[du=(4x^3+6x)\,dx=2(2x^3+3x)\,dx \nonumber\]

\[\dfrac{1}{2}\,du=(2x^3+3x)\,dx. \nonumber\]

Rewrite the integrand in \(u\):

\[∫\dfrac{2x^3+3x}{x^4+3x^2}\,dx=\dfrac{1}{2}∫\frac{1}{u}\,du. \nonumber\]

Then we have

\[\dfrac{1}{2}∫\frac{1}{u}\,du=\dfrac{1}{2}\ln |u|+C=\dfrac{1}{2}\ln ∣x^4+3x^2∣+C. \nonumber\]

Example \(\PageIndex{11}\) is a definite integral of a trigonometric function. With trigonometric functions, we often have to apply a trigonometric property or an identity before we can move forward. Finding the right form of the integrand is usually the key to a smooth integration.

Example \(\PageIndex{11}\): Evaluating a Definite Integral

Evaluate the definite integral \[∫^{π/2}_0\dfrac{\sin x}{1+\cos x}\,dx.\nonumber\]

**Solution**

We need substitution to evaluate this problem. Let \(u=1+\cos x\) so \(du=−\sin x\,\,dx.\)

Rewrite the integral in terms of \(u\), changing the limits of integration as well. Thus,

\[ \begin{align*} u &= 1+\cos(0)=2 \\[4pt] u &=1+\cos \left(\dfrac{π}{2}\right)=1.\end{align*}\]

Then

\[ \begin{align*}∫^{π/2}_0\dfrac{\sin x}{1+\cos x} &=−∫^1_2 \frac{1}{u}\,du \\[4pt] &=∫^2_1\frac{1}{u}\,du \\[4pt] &=\ln |u|\,\bigg|^2_1 \\[4pt] &=[\ln 2−\ln 1]=\ln 2 \end{align*}\]

## Key Concepts

- Exponential and logarithmic functions arise in many real-world applications, especially those involving growth and decay.
- Substitution is often used to evaluate integrals involving exponential functions or logarithms.

## Key Equations

**Integrals of Exponential Functions**

\[∫e^x\,dx=e^x+C \nonumber\]

\[\int a^x\,dx=\dfrac{a^x}{\ln a}+C \nonumber\]

**Integration Formulas Involving Logarithmic Functions**

\[∫\frac{1}{x}\,dx=\ln |x|+C \nonumber\]

\[∫\frac{u'(x)}{u(x)}\,dx = \ln |u(x)|+C \nonumber\]

## Contributors and Attributions

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

- Paul Seeburger (Monroe Community College) substantially edited the subsection titled, Integration Resulting in Natural Logarithms.