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.

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

Try It

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

Hint

Follow the pattern from the last example to solve the problem.

Show Solution

[latex]\text{ln}|x+2|+C[/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]

Show Solution

This can be rewritten as [latex]\displaystyle\int (2{x}^{3}+3x){({x}^{4}+3{x}^{2})}^{-1}dx.[/latex] Use substitution. Let [latex]u={x}^{4}+3{x}^{2},[/latex] then [latex]du=4{x}^{3}+6x.[/latex] Alter du by factoring out the 2. Thus,

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.

Evaluating a Definite Integral

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

Show Solution

We need substitution to evaluate this problem. Let [latex]u=1+ \cos x,,[/latex] so [latex]du=\text{−} \sin xdx.[/latex] Rewrite the integral in terms of [latex]u[/latex], changing the limits of integration as well. Thus,