Question

In the following exercises, find each indefinite integral, using appropriate substitutions. \(\int \frac{d x}{|x| \sqrt{x^{2}-9}}\)

Short answer

The indefinite integral is \(\frac{1}{9}\sec^{-1}\left(\frac{x}{3}\right) + C\).

Step-by-step solution

Recognize the Type of Integral

The given integral involves a square root of the form \(x^2 - a^2\), suggesting hyperbolic substitution might be useful. Specifically, the form \(\sqrt{x^2 - 9}\) can be approached using a substitution involving hyperbolic functions.

Choose the Substitution

Set \(x = 3\sec(\theta)\). This implies \(|x| = 3|\sec(\theta)|\) and \(dx = 3\sec(\theta)\tan(\theta) d\theta\). We choose this substitution because \(\sec^2(\theta) - 1 = \tan^2(\theta)\), which simplifies the expression under the square root.

Substitute and Simplify the Integral

Substitute \(x = 3\sec(\theta)\) in the integral:\[\int \frac{1}{|3\sec(\theta)| \sqrt{(3\sec(\theta))^2 - 9}} \cdot 3\sec(\theta)\tan(\theta) d\theta\]Simplifying, this becomes:\[\int \frac{3\sec(\theta)\tan(\theta)}{3|\sec(\theta)| \cdot 3\tan(\theta)} d\theta = \int \frac{1}{9} d\theta\]This results because:\((3\sec(\theta))^2 - 9 = 9\tan^2(\theta)\), simplifying \(\sqrt{9\tan^2(\theta)} = 3|\tan(\theta)|\).

Perform the Integration

The integral simplifies to:\[\int \frac{1}{9} d\theta = \frac{1}{9}\theta + C\]where \(C\) is the constant of integration.

Back-substitute to Original Variable

Recall the substitution \(x = 3\sec(\theta)\), which implies \(\sec(\theta) = \frac{x}{3}\). Therefore, \(\theta = \sec^{-1}\left(\frac{x}{3}\right)\). Substituting back gives:\[\int \frac{1}{|x| \sqrt{x^2 - 9}} dx = \frac{1}{9} \sec^{-1}\left(\frac{x}{3}\right) + C\]

Key concepts

Hyperbolic Substitution

Hyperbolic substitution is a technique in calculus used to simplify integrals that involve square roots of expressions in the form \(x^2 - a^2\). This method is particularly useful because the identity of hyperbolic functions, similar to trigonometric functions, help us in transforming the integral into a more manageable form.

When we see integrals with \(\sqrt{x^2 - a^2}\), like the one in our problem \(\sqrt{x^2 - 9}\), hyperbolic substitution becomes a valuable tool. The idea is to substitute \(x\) with a hyperbolic function, such as \(x = a \sec(\theta)\), leveraging the relationship between hyperbolic functions.

The core identity that assists us here is:

• \( \sec^2(\theta) - 1 = \tan^2(\theta) \)

This transformation simplifies the square root expression into a form without radicals, making it easier to integrate. Through hyperbolic substitution, the integral simplifies significantly, removing complex terms and allowing basic algebraic operations to solve them easily.

Integral Calculus

Integral calculus is a major branch of calculus that focuses on the process of integration, which can be thought of as finding the whole from known parts. It deals with the accumulation of quantities, such as areas under a curve or the total distance traveled by a moving object.

In the context of this exercise, we are dealing with indefinite integrals, which involve finding the anti-derivative of a given function. This means, instead of evaluating the exact numerical value, we look for an expression that, when differentiated, yields the integrand function.

The indefinite integral is represented by:

• \(\int f(x) \, dx\)

Here, \(f(x)\) is the function to be integrated, and \(dx\) indicates a small change in \(x\). The solution, or the anti-derivative, includes an arbitrary constant \(C\), accounting for all possible vertical shifts in the anti-derivative function. Integral calculus applies various methods and substitutions to solve complex integrals, transforming them into simpler expressions for easy evaluation.

Substitution Method

The substitution method, also called the method of u-substitution, is a common technique in integration used to simplify problems by changing variables. It can be especially helpful in dealing with complex integrals that seem challenging at first glance.

In essence, the substitution method transforms a difficult integral into an easier form, making it possible to apply basic integration formulas. With this technique, you substitute part of the original integral by a new variable, say \(u\), which streamlines the expression to a familiar form. The steps involved in the substitution method include:

• Identifying a substitution \(u = g(x)\)
• Calculating \(du = g'(x) \, dx\)
• Transforming the integral with respect to \(u\)
• Performing the integration in terms of \(u\)
• Back-substituting \(u\) to express the result in the original variable \(x\)

In the exercise, we substituted \(x = 3\sec(\theta)\), which transformed the integral into terms of \(\theta\). After simplifying and integrating, back-substitution allowed us to express the final solution in terms of the original variable \(x\). This step-by-step transformation is what makes substitution a powerful strategy in integral calculus.