## Integrals Involving Logarithmic Functions

### Learning Outcomes

• Integrate functions involving logarithmic functions

Integrating functions of the form $f(x)={x}^{-1}$ result in the absolute value of the natural log function, as shown in the following rule. Integral formulas for other logarithmic functions, such as $f(x)=\text{ln}x$ and $f(x)={\text{log}}_{a}x,$ are also included in the rule.

### Integration Formulas Involving Logarithmic Functions

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

$\begin{array}{ccc}\hfill \displaystyle\int {x}^{-1}dx& =\hfill & \text{ln}|x|+C\hfill \\ \hfill \displaystyle\int \text{ln}xdx& =\hfill & x\text{ln}x-x+C=x(\text{ln}x-1)+C\hfill \\ \hfill \displaystyle\int {\text{log}}_{a}xdx& =\hfill & \frac{x}{\text{ln}a}(\text{ln}x-1)+C\hfill \end{array}$

### Example: Finding an Antiderivative Involving $\text{ln}x$

Find the antiderivative of the function $\dfrac{3}{x-10}.$

### Try It

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

### Example: Finding an Antiderivative of a Rational Function

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

Watch the following video to see the worked solution to Example: Finding an Antiderivative of a Rational Function.

### Example: Finding an Antiderivative of a Logarithmic Function

Find the antiderivative of the log function ${\text{log}}_{2}x.$

### Try It

Find the antiderivative of ${\text{log}}_{3}x.$

Watch the following video to see the worked solution to the above Try It.

The example below 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.

### Evaluating a Definite Integral

Find the definite integral of ${\displaystyle\int }_{0}^{\pi \text{/}2}\frac{ \sin x}{1+ \cos x}dx.$