Problem 53 - Calculus Early Transcendentals

Question

Determine the following indefinite integrals. Check your work by differentiation. $$\int \frac{1}{x \sqrt{x^{2}-25}} d x$$

Step-by-Step Solution

Answer

Question: Evaluate the indefinite integral of the function \(f(x) = \frac{1}{x\sqrt{x^2 - 25}}\). Answer: The indefinite integral of the given function is \(\int f(x) dx = \frac{1}{5}\operatorname{sec}^{-1}(\frac{x}{5}) + C\).

Step 1: Identify the appropriate substitution

The given integral has the form \(\int \frac{1}{x\sqrt{x^2 - 25}} \, dx\). Recognize that the denominator has a square root term with \(x^2 - 25\), which suggests a trigonometric substitution. We'll use the secant substitution \(x = 5\operatorname{sec}(\theta)\). Also, find the derivative of x in terms of θ: \(dx = 5\operatorname{sec}(\theta)\operatorname{tan}(\theta)d\theta\).

Step 2: Apply substitution and transform the integral

Substitute \(x = 5\operatorname{sec}(\theta)\) and \(dx = 5\operatorname{sec}(\theta)\operatorname{tan}(\theta)d\theta\) to transform the original integral: $$ \int{\frac{1}{5\operatorname{sec}(\theta)\sqrt{(5\operatorname{sec}(\theta))^2 - 25}} \cdot 5\operatorname{sec}(\theta)\operatorname{tan}(\theta)d\theta} $$

Step 3: Evaluate the transformed integral

Simplify the integral: $$ \int{\frac{5\operatorname{sec}(\theta)\operatorname{tan}(\theta)d\theta}{5\operatorname{sec}(\theta)\sqrt{25\operatorname{sec}^2(\theta) - 25}}} $$ Cancel the 5's and sec terms and simplify further: $$ \int{\frac{\operatorname{tan}(\theta)d\theta}{\sqrt{25(\sec^2(\theta) - 1)}}} $$ Now using the trigonometric identity \(\sec^2(\theta) - 1 = \tan^2(\theta)\), we have: $$ \int{\frac{\operatorname{tan}(\theta)d\theta}{5\operatorname{tan}(\theta)}} $$ Cancel the tan terms: $$ \int{\frac{1}{5}d\theta} $$

Step 4: Reverse the substitution to get the integral in terms of x

The integral now becomes: $$ \frac{1}{5}\int{1\,d\theta} = \frac{1}{5}\theta + C $$ Now we need to convert back to x. Recall that \(x = 5\operatorname{sec}(\theta)\). Solving for \(\theta\) gives: $$ \theta = \operatorname{sec}^{-1}(\frac{x}{5}) $$ So, the integral simplifies to: $$ \frac{1}{5}\operatorname{sec}^{-1}(\frac{x}{5}) + C $$

Step 5: Differentiate the result to verify

Differentiate the computed integral with respect to x to verify the result: $$ \frac{d}{dx} \left(\frac{1}{5}\operatorname{sec}^{-1}(\frac{x}{5}) + C\right) = \frac{1}{5}\frac{1}{\sqrt{1-\frac{x^2}{25}}} \cdot \frac{d}{dx}\left(\frac{x}{5}\right) $$ Simplifying further gives the original integrand: $$ \frac{1}{x\sqrt{x^2-25}} $$ Verification is successful. The integral of the given function is: $$ \int \frac{1}{x \sqrt{x^2-25}} dx = \frac{1}{5}\operatorname{sec}^{-1}(\frac{x}{5}) + C $$

Key Concepts

Trigonometric Substitution

When faced with integrals involving expressions like \( \sqrt{x^2 - 25} \), trigonometric substitution is a powerful tool. This technique transforms algebraic expressions into trigonometric ones, making them easier to integrate.
To decide on the appropriate substitution, examine the expression under the square root. Here, \/( x^2 - 25 \/) suggests using the identity \( \sec^2(\theta) - 1 = \tan^2(\theta) \).
This identity helps simplify the integral by rewriting \( x = 5\sec(\theta) \).

• The choice \( x = 5\sec(\theta) \) transforms \( \sqrt{x^2 - 25} \) into \( 5\tan(\theta) \), using the trigonometric identity.
• Then, compute \( dx \) as the derivative \( 5\sec(\theta)\tan(\theta)d\theta \).

This substitution converts the original algebraic integrand into a simpler form involving only trigonometric functions, which can be straightforwardly integrated.

Differentiation Verification

After integrating an expression, it is crucial to verify the result by differentiation. This step confirms the correctness of the integration process.
Differentiate the result \( \frac{1}{5}\sec^{-1}(\frac{x}{5}) + C \) with respect to \( x \). Use the chain rule and known derivative formulas.

• The derivative of \( \sec^{-1}(x) \) is \( \frac{1}{x\sqrt{x^2 - 1}} \).
• Apply the chain rule: \( \frac{d}{dx}\left(\sec^{-1}(\frac{x}{5})\right) = \frac{1}{\sqrt{1 - \left(\frac{x}{5}\right)^2}} \cdot \frac{1}{5} \).

By simplifying, the differentiated form \( \frac{1}{x\sqrt{x^2 - 25}} \) should match the original integrand.
Successful matching confirms that the integration was performed accurately.

Integration Techniques

Understanding a variety of integration techniques allows for tackling different kinds of integrals. Trigonometric substitution is one among many methods.
Different integrals require distinct approaches, and some of the frequently used techniques include:

• Substitution: Replacing a part of the integrand with a single variable to simplify the integration process.
• Integration by Parts: Useful for products of functions, derived from the product rule of differentiation.
• Partial Fraction Decomposition: Used for rational functions, dividing them into simpler fractions that can be integrated easily.

Choosing the most effective technique often depends on recognizing patterns and understanding the structure of the integrand.
For the problem at hand, trigonometric substitution was optimal due to the expression \( x^2 - 25 \), allowing simplification through known trigonometric identities.