Question

Evaluate the following definite integrals. Use Theorem 7.7 to express your answer in terms of logarithms. $$\int_{5}^{3 \sqrt{5}} \frac{d x}{\sqrt{x^{2}-9}}$$

Short answer

Question: Evaluate the definite integral $$\int_{5}^{3 \sqrt{5}} \frac{d x}{\sqrt{x^{2}-9}}$$ and express the result in terms of logarithms using Theorem 7.7. Answer: $$\int_{5}^{3 \sqrt{5}} \frac{d x}{\sqrt{x^{2}-9}} = \ln{\frac{\sec{(\frac{\pi}{2})} + \tan{(\frac{\pi}{2})}}{\sec{(\frac{\pi}{3})} + \tan{(\frac{\pi}{3})}}}$$

Step-by-step solution

Identify trigonometric substitution

We see that the integral has a square root of a quadratic form: \(\sqrt{x^2 - 9}\). To tackle this, we can use the following trigonometric substitution: $$x = 3 \sec{(\theta)},$$ because when we do the substitution, we'll get a simple expression inside the square root.

Differentiate x with respect to θ

Find dx/dθ to substitute dx in the integral: $$\frac{dx}{d\theta} = 3\sec{(\theta)}\tan{(\theta)}$$

Substitute x and dx in the integral

Replace x and dx in the integral with the corresponding expressions: $$\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{3\sec{(\theta)}\tan{(\theta)}d\theta}{\sqrt{9\sec^2{(\theta)}-9}}$$

Simplify the integrand

We notice that the expression inside the square root simplifies nicely: $$\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{3\sec{(\theta)}\tan{(\theta)}d\theta}{\sqrt{9\tan^2{(\theta)}}}$$ $$= \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{3\sec{(\theta)}\tan{(\theta)}d\theta}{3\tan{(\theta)}}$$ $$= \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \sec{(\theta)} d\theta$$

Integrate sec(theta)

The integral of sec(θ) is well-known: $$\int \sec{(\theta)} d\theta = \ln|\sec{(\theta)} + \tan{(\theta)}| + C$$

Evaluate the result over the interval

Now, we just need to evaluate this result for the given interval: $$\ln|\sec{(\frac{\pi}{2})} + \tan{(\frac{\pi}{2})}| - \ln|\sec{(\frac{\pi}{3})} + \tan{(\frac{\pi}{3})}|$$

Final result

Lastly, express the result in terms of logarithms using Theorem 7.7: $$\int_{5}^{3 \sqrt{5}} \frac{d x}{\sqrt{x^{2}-9}} = \ln{\frac{\sec{(\frac{\pi}{2})} + \tan{(\frac{\pi}{2})}}{\sec{(\frac{\pi}{3})} + \tan{(\frac{\pi}{3})}}}$$

Key concepts

Trigonometric Substitution

Trigonometric substitution is a powerful technique that simplifies the integration of expressions containing square roots of quadratic equations. It's particularly useful when dealing with integrals of the form \( \sqrt{a^2 - x^2} \), \( \sqrt{x^2 - a^2} \), or \( \sqrt{x^2 + a^2} \). The method involves replacing the variable \( x \) with a trigonometric function that simplifies the expression under the square root.

In this problem, we faced an integral with \( \sqrt{x^2 - 9} \), which fits the pattern \( \sqrt{x^2 - a^2} \). Here, a good choice for substitution is \( x = 3 \sec(\theta) \). This substitution turns the square root \( \sqrt{x^2 - 9} = \sqrt{9 \sec^2(\theta) - 9} \) into \( 3 \tan(\theta) \), making the integral easier to manage.

After substitution, we also need to replace \( dx \) using the derivative of the substitution:

• \( \frac{dx}{d\theta} = 3 \sec(\theta) \tan(\theta) \)

This allows us to express the integral entirely in terms of \( \theta \), facilitating the process towards evaluating the definite integral.

Integration Techniques

Integration techniques are essential tools in calculus that help to find antiderivatives or integrals of functions. These techniques range from basic rules to more complex methods. In this context, trigonometric substitution is one such method used to rewrite the integral into a more manageable form by using trigonometric identities.

After the trigonometric substitution \( x = 3 \sec(\theta) \) and expressing \( dx \) in terms of \( \theta \), we were left with the integral:
\[ \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \sec(\theta) d\theta \]
To solve this, we needed to integrate \( \sec(\theta) \), a standard but less intuitive form. The integral \( \int \sec(\theta) d\theta \) is known to be:

• \( \ln|\sec(\theta) + \tan(\theta)| + C \)

Various integration techniques like substitution, partial fractions, and integration by parts are employed depending on the function's composition. In this scenario, recognizing that \( \sec \) has a straightforward integral form aided in evaluating the definite integral across the interval.

Logarithmic Functions

Logarithmic functions often appear in the results of certain integrals, like those involving secant functions. They provide a concise way to express complex relationships, typical in calculus problems.

In our exercise, after integrating \( \sec(\theta) \), the result was:
\[ \ln|\sec(\theta) + \tan(\theta)| + C \]
This expression highlights the logarithmic function's role in simplifying the integration of trigonometric forms. The last step involved evaluating this expression from \( \frac{\pi}{3} \) to \( \frac{\pi}{2} \).

Finally, by substituting back the original variable and calculating the values:

• \( \ln|\sec(\frac{\pi}{2}) + \tan(\frac{\pi}{2})| \)
• \( \ln|\sec(\frac{\pi}{3}) + \tan(\frac{\pi}{3})| \)

We obtained the result in a neat logarithmic form:
\[ \ln{\frac{\sec(\frac{\pi}{2}) + \tan(\frac{\pi}{2})}{\sec(\frac{\pi}{3}) + \tan(\frac{\pi}{3})}} \]
This logarithmic expression efficiently captures the complex nature of the antiderivative over the given interval. Logarithms are a quintessential part of integration that often simplify not only the computations but also the understanding of an integral's behavior.