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.

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

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

Show Solution

First factor the 3 outside the integral symbol. Then use the [latex]u^{-1}[/latex] rule. Thus,

[latex]\begin{array}{ll}{\displaystyle\int \frac{3}{x-10}dx}\hfill & =3{\displaystyle\int \frac{1}{x-10}dx}\hfill \\ \\ \\ & =3{\displaystyle\int \frac{du}{u}}\hfill \\ & =3\text{ln}|u|+C\hfill \\ & =3\text{ln}|x-10|+C,x\ne 10.\hfill \end{array}[/latex]

See Figure 3.

Figure 3. The domain of this function is [latex]x\ne 10.[/latex]

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

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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.