Question

In the following exercises, find each indefinite integral, using appropriate substitutions. \(\int \frac{d x}{|x| \sqrt{4 x^{2}-16}}\)

Short answer

The indefinite integral is \(\frac{1}{2} \sec^{-1}\left(\frac{x}{2}\right) + C\).

Step-by-step solution

Identify the Substitution

First, observe the expression inside the square root, which is \(4x^2 - 16\). Notice that it can be factored as \(4(x^2 - 4)\). This suggests using a trigonometric substitution. Recognize \(x^2 - 4\) resembles the form \(a^2 - u^2\) where \(u = x\) and \(a = 2\). Let's use the substitution \(x = 2\sec\theta\).

Perform the Substitution

With the substitution \(x = 2\sec\theta\), we find \(dx = 2\sec\theta \tan\theta d\theta\). Substitute these into the integral:\[\int \frac{d x}{|2\sec\theta| \sqrt{4(2\sec\theta)^2 - 16}} = \int \frac{2\sec\theta \tan\theta d\theta}{2|\sec\theta| \cdot 2\sqrt{ \sec^2\theta - 1}}\]

Simplify Under the Root and in the Denominator

Recall \(\sec^2\theta - 1 = \tan^2\theta\), thus \(\sqrt{\sec^2\theta - 1} = \tan\theta\). The integral becomes:\[\int \frac{2\sec\theta \tan\theta d\theta}{2|\sec\theta| \cdot 2 \tan\theta} \]Simplify the expression by canceling out \(\tan\theta\) and some constant terms. The integral becomes:\[\int \frac{1}{2} d\theta\]

Integrate with respect to θ

The integral \(\int \frac{1}{2} d\theta\) is straightforward. Integrate to obtain:\[\frac{1}{2}\theta + C\]where \(C\) is the constant of integration.

Back-substitute to x

Return to the original variable \(x\) where \(x = 2\sec\theta\). Rearranging gives \(\sec\theta = \frac{x}{2}\) leading to \(\theta = \sec^{-1}\left(\frac{x}{2}\right)\). Substitute back into the integral result:\[\frac{1}{2}\left(\sec^{-1}\left(\frac{x}{2}\right)\right) + C\]

Key concepts

Trigonometric Substitution

In calculus, trigonometric substitution is a powerful technique used to evaluate integrals involving expressions like \(a^2 - x^2\), \(a^2 + x^2\), or \(x^2 - a^2\). These expressions can be simplified by substituting a trigonometric identity to covert the integral into a more manageable form. For example, when you see expressions like \(\sqrt{4x^2 - 16}\), it's a signal that trigonometric substitution might be the right path.

In our case, we identified that \(4x^2 - 16\) can be factored into \(4(x^2 - 4)\), which resembles the form \(a^2 - x^2\). By using the substitution \(x = 2\sec\theta\), we transform the integral into trigonometric terms. This step reduces the complexity of the problem and is a crucial step in solving it efficiently.

Trigonometric substitution turns the imitation of radicals within the integrand into simpler trigonometric functions, making calculations more straightforward. This substitution is not only helpful for solving integrals but also enhances understanding of relationships between algebraic and trigonometric forms.

Integration Techniques

Integration techniques are varied methods used to solve integrals that cannot be computed directly. One such technique is substitution, both algebraic and trigonometric. In this exercise, we focus on trigonometric substitution, which involves transforming the variable into a trigonometric function to simplify the integral.

After performing the trigonometric substitution of \(x = 2\sec\theta\) in our integral, we simplify it by recalling identities such as \(\sec^2\theta - 1 = \tan^2\theta\). This substitution helps replace complex expressions with simpler trigonometric ones. Once the substitution is complete, simplifying the integral using trigonometric identities is the next step.

By converting the problem into trigonometric terms, we allowed the integral to reduce to a much easier form \(\int \frac{1}{2} d\theta\), greatly simplifying the process. Mastering different integration techniques, like substitution, is vital for systematically solving calculus problems.

Calculus Problems

Calculus problems often challenge students to apply different concepts and techniques learned throughout their studies. By flexibly combining methods like trigonometric substitution with strong grounding in derivative and integral concepts, solving complex integrals becomes more achievable.

The original problem \(\int \frac{d x}{|x| \sqrt{4 x^{2}-16}}\) required recognizing when a substitute, such as a trigonometric function, could simplify an intimidating expression. This is a common scenario in calculus where one identifies the form of the expression and chooses an appropriate method to tackle it.

Whether differentiating or integrating, understanding when and why to use a particular technique is crucial. Problems like these test a student's ability to see beyond the numbers and symbols, challenging them to think critically and creatively about the best approach. Successfully solving such calculus problems involves not only applying known techniques but also recognizing patterns and finding simplifications.