Problem 74 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 73-76: Make the trigonometric substitution $$ x=a \sec \theta \text { for } 0<\theta<\pi / 2 \text { and } a>0 \text {. } $$ Simplify the resulting expression. $$ \frac{1}{x^{2} \sqrt{x^{2}-a^{2}}} $$

Step-by-Step Solution

Answer

\(\frac{\cos^3 \theta}{a^3 \sin \theta}\)

Step 1: Substitute for x

Given the substitution \( x = a \sec\theta \), replace \( x \) in the expression with \( a \sec\theta \). The new expression becomes:\[ \frac{1}{(a \sec \theta)^2 \sqrt{(a \sec \theta)^2 - a^2}} \]

Step 2: Simplify the Expression

First, simplify \( (a \sec \theta)^2 \):\[ (a \sec \theta)^2 = a^2 \sec^2 \theta \]Substitute back into the expression:\[ \frac{1}{a^2 \sec^2 \theta \sqrt{a^2 \sec^2 \theta - a^2}} \]

Step 3: Factor and Simplify the Square Root

Factor out \( a^2 \) from the square root:\[ \sqrt{a^2 (\sec^2 \theta - 1)} = a \sqrt{\sec^2 \theta - 1} \]Recognize that \( \sec^2 \theta - 1 = \tan^2 \theta \), so:\[ a \sqrt{\sec^2 \theta - 1} = a \tan \theta \]Now, substitute back into the expression:\[ \frac{1}{a^2 \sec^2 \theta \cdot a \tan \theta} \]

Step 4: Final Simplification

The expression further simplifies by canceling terms:\[ \frac{1}{a^3 \sec^2 \theta \tan \theta} \]Express \( \sec \theta \) and \( \tan \theta \) in terms of sine and cosine to further simplify:\[ \sec \theta = \frac{1}{\cos \theta}, \quad \tan \theta = \frac{\sin \theta}{\cos \theta} \]Thus:\[ \frac{1}{a^3 \left( \frac{1}{\cos^2 \theta} \right) \left( \frac{\sin \theta}{\cos \theta} \right)} = \frac{\cos^3 \theta}{a^3 \sin \theta} \]

Step 5: Simplified Result

The expression has been simplified to:\[ \frac{\cos^3 \theta}{a^3 \sin \theta} \]

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical tools that relate the angles and sides of triangles. They are especially useful when simplifying expressions or solving equations involving trigonometric functions. In our exercise, the identity for the secant function is key. It is given by:

• \[ \sec \theta = \frac{1}{\cos \theta} \]
• The identity relating secant and tangent is:\[ \sec^2 \theta - 1 = \tan^2 \theta \]

These identities are powerful because they allow us to replace complex-looking expressions with simpler ones. In the substitution and simplification process, recognizing that \( \sec^2 \theta - 1 = \tan^2 \theta \) helped to simplify the square root in the problem. By substituting \( \sec \theta \) with \( \frac{1}{\cos \theta} \) and \( \tan \theta \) with \( \frac{\sin \theta}{\cos \theta} \), the problem becomes much more manageable.

Simplification Techniques

Simplifying expressions is an essential skill in mathematics, especially in calculus, where expressions can become complex. The primary goal is to reduce an expression to its most basic form. Let's go through the simplifications in our exercise.Initially, after substituting \(x = a \sec \theta\), the expression became more complex. However, seeing \(a^2(\sec^2 \theta - 1)\) inside the square root allows us to recognize a chance to simplify. By factoring out \(a^2\), and using the trigonometric identity \(\sec^2 \theta - 1 = \tan^2 \theta\), we transform the expression to involve \(\tan \theta\), a simpler function.The fraction further simplifies by substituting trigonometric functions with their definitions in terms of sine and cosine:

• \(\sec \theta = \frac{1}{\cos \theta}\)
• \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)

This ultimately yields \(\frac{\cos^3 \theta}{a^3 \sin \theta}\), removing the square and radical operations from the initial expression.

Calculus Integration Methods

Integrating functions, especially those involving square roots and trigonometric expressions, can be challenging. One effective approach is using trigonometric substitution, which simplifies the integral by transforming variables. In this exercise, we used the substitution \(x = a \sec \theta\) to handle a term with \(\sqrt{x^2-a^2}\).Trigonometric substitution is particularly useful when:

• You encounter expressions of the form \(\sqrt{x^2 - a^2}\).
• Substituting with a trigonometric function can lead to simpler integrals that are easier to integrate.

The substitution aligns perfectly with the identity \(\sec^2 \theta - 1 = \tan^2 \theta\), thereby simplifying the square root term.After substitution, new expressions in terms of \(\theta\) often emerge, allowing for easier integration. They also frequently reduce integrals involving square roots to basic trigonometric integral forms, which are well-documented and easier to solve.