Problem 70 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 69-72: Make the trigonometric substitution $$ x=a \tan \theta \text { for }-\pi / 2<\theta<\pi / 2 \text { and } a>0 \text {. } $$ Simplify the resulting expression. $$ \frac{1}{\sqrt{a^{2}+x^{2}}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{\cos \theta}{a} \).

Step 1: Identify the Substitution

The problem suggests using the substitution \( x = a \tan \theta \). This substitution is based on trigonometric identities and helps in simplifying expressions involving \( x^2 + a^2 \).

Step 2: Substitute for x in the Expression

Replace \( x \) in the expression \( \frac{1}{\sqrt{a^2 + x^2}} \) with \( a \tan \theta \). This gives us the expression \( \frac{1}{\sqrt{a^2 + (a \tan \theta)^2}} \).

Step 3: Simplify Inside the Square Root

Simplify the expression inside the square root: \( a^2 + (a \tan \theta)^2 = a^2 + a^2 \tan^2 \theta = a^2(1 + \tan^2 \theta) \).

Step 4: Use Trigonometric Identity

Recall the identity \( 1 + \tan^2 \theta = \sec^2 \theta \). Substitute this into the expression to get \( \sqrt{a^2 \sec^2 \theta} = a \sec \theta \).

Step 5: Simplify the Expression

Substitute the simplified square root back into the original expression: \( \frac{1}{\sqrt{a^2 + x^2}} = \frac{1}{a \sec \theta} \). Further simplifying gives \( \frac{1}{a} \cos \theta \).

Step 6: Final Expression

The expression simplifies to \( \frac{\cos \theta}{a} \), which is the result after applying the trigonometric substitution and simplifying.

Key Concepts

Trigonometric Identities

Trigonometric identities are formulas that express relationships between the angles and sides of triangles. They provide tools that help us simplify expressions, especially in integral calculus and trigonometry.
One of the key identities useful in this exercise is the Pythagorean identity:

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

This identity helps us to substitute and simplify expressions when dealing with square roots formed by \(x^2 + a^2\).
In this exercise, the substitution \(x = a \tan \theta\) creates an expression \(a^2(1 + \tan^2 \theta)\) inside the square root. By recognizing the trigonometric identity \(1 + \tan^2 \theta = \sec^2 \theta\), we can further simplify it to \(a^2 \sec^2 \theta\).
Thus, using trigonometric identities effectively allows us to transform complicated algebraic expressions into simpler, more manageable forms.

Simplifying Expressions

Simplifying expressions involves rewriting them in the simplest possible form. This is often necessary for ease of computation, especially when performing integration or solving more complex math problems.
In our problem, the substitution \(x = a \tan \theta\) changes the initial expression to \(\frac{1}{\sqrt{a^2 + (a \tan \theta)^2}}\). Through substitution, the goal is to express the problem using simpler terms based on known trigonometric identities and properties.
After substitution, the expression inside the square root becomes \(a^2 (1 + \tan^2 \theta)\). Applying the identity \(1 + \tan^2 \theta = \sec^2 \theta\), we simplify it further to \(\sqrt{a^2 \sec^2 \theta}\).
Finally, simplifying gives us \(a \sec \theta\), allowing us to substitute back into the original expression to obtain \(\frac{1}{a \sec \theta} = \frac{\cos \theta}{a}\).
Throughout this process, each step aims to break down and reduce complexity, transforming the expression into its most straightforward form.

Integration Techniques

Integration techniques are strategies used to find integrals of functions that cannot be easily integrated using basic methods. Trigonometric substitution is one such technique that leverages trigonometric identities to simplify integrals that involve square roots.
The problem involves integrating expressions like \(\sqrt{a^2 + x^2}\), where direct integration isn’t straightforward. By using the trigonometric substitution \(x = a \tan \theta\), the task becomes manageable by reducing it to simpler trigonometric forms.
This substitution alters the integral and simplifies things by transforming square root expressions into constants and trigonometric identities, greatly facilitating the evaluation of integrals.
Using \(x = a \tan \theta\), the integrand becomes \(\frac{1}{a \sec \theta}\), which is simpler than the original square root form. This form is easier to integrate because \(\sec \theta\) and associated trigonometric functions lend themselves well to standard integration techniques.
Overall, integration techniques like trigonometric substitution are powerful tools that convert challenging problems into solvable forms, making them essential for complex calculus tasks.