Integrals Involving Logarithmic Functions

Learning Outcomes

  • Integrate functions involving logarithmic functions

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

Integration Formulas Involving Logarithmic Functions


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

[latex]\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}[/latex]

 

Example: Finding an Antiderivative Involving [latex]\text{ln}x[/latex]

Find the antiderivative of the function [latex]\dfrac{3}{x-10}.[/latex]

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Find the antiderivative of [latex]\dfrac{1}{x+2}.[/latex]

Example: Finding an Antiderivative of a Rational Function

Find the antiderivative of [latex]\dfrac{2{x}^{3}+3x}{{x}^{4}+3{x}^{2}}.[/latex]

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 [latex]{\text{log}}_{2}x.[/latex]

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Find the antiderivative of [latex]{\text{log}}_{3}x.[/latex]

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 [latex]{\displaystyle\int }_{0}^{\pi \text{/}2}\frac{ \sin x}{1+ \cos x}dx.[/latex]

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