Question

Use a CAS to evaluate the following integrals. Tables can also be used to verify the answers. $$ \int \frac{\sqrt{x^{2}-9}}{3 x} d x $$

Short answer

The integral is \( \frac{\sqrt{x^2 - 9}}{3} - \sec^{-1}\left(\frac{x}{3}\right) + C \).

Step-by-step solution

Identify the Integral Type

The given integral is \( \int \frac{\sqrt{x^{2}-9}}{3x} \ dx \). This integral involves a square root of a quadratic expression, which might hint at using a trigonometric substitution method.

Choose Appropriate Substitution

Since the integral involves \( \sqrt{x^2 - 9} \), a suitable substitution is \( x = 3\sec(\theta) \). This simplifies \( \sqrt{x^2 - 9} = 3\tan(\theta) \) and \( dx = 3\sec(\theta)\tan(\theta) \ d\theta \).

Substitute and Simplify

Substitute \( x = 3\sec(\theta) \) into the integral:\[\int \frac{3\tan(\theta)}{3 \cdot 3\sec(\theta)} \cdot 3\sec(\theta)\tan(\theta) \ d\theta.\]Simplify the expression:\[\int \tan^2(\theta) \ d\theta.\]

Solve the Integral

Using the identity \( \tan^2(\theta) = \sec^2(\theta) - 1 \), we rewrite the integral:\[\int (\sec^2(\theta) - 1) \ d\theta = \int \sec^2(\theta) \ d\theta - \int d\theta.\]This evaluates to:\( \tan(\theta) - \theta + C \).

Convert Back to Original Variable

Substitute back \( \theta \) in terms of \( x \) using \( \tan(\theta) = \frac{\sqrt{x^2 - 9}}{3} \) and \( \theta = \sec^{-1}\left(\frac{x}{3}\right) \):\[\frac{\sqrt{x^2 - 9}}{3} - \sec^{-1}\left(\frac{x}{3}\right) + C.\]

Verify with a CAS

Verify this solution using a Computer Algebra System (CAS) such as Wolfram Alpha or a graphing calculator to ensure the integral evaluation is correct.

Key concepts

Trigonometric Substitution

Trigonometric substitution is a technique used in calculus to simplify the integration of expressions containing square roots of quadratic polynomials. It's especially useful when the integrand involves terms like \( \sqrt{a^2 - x^2} \), \( \sqrt{x^2 - a^2} \), or \( \sqrt{x^2 + a^2} \). In these cases, we use trigonometric identities to transform the integrand into a simpler form that is easier to integrate.

The key idea is to substitute the part of the expression under the square root with a trigonometric identity. For instance:

• For \( \sqrt{a^2 - x^2} \), use \( x = a\sin(\theta) \).
• For \( \sqrt{x^2 - a^2} \), use \( x = a\sec(\theta) \).
• For \( \sqrt{x^2 + a^2} \), use \( x = a\tan(\theta) \).

By changing variables, the problem reduces to simpler trigonometric integrals. After solving the integral, don't forget to substitute back to the original variable to get the final answer.

In the exercise, we used \( x = 3\sec(\theta) \) for the integral \( \int \frac{\sqrt{x^{2}-9}}{3x} \, dx \), because the expression \( \sqrt{x^2 - 9} \) fits the form \( \sqrt{x^2 - a^2} \). The substitution transforms it into \( 3\tan(\theta) \), simplifying the integration process.

Definite Integrals

Definite integrals are used to calculate the area under a curve over a specific interval. This contrasts with indefinite integrals, which represent an antiderivative or a family of functions. When solving a definite integral, we compute the net area, allowing us to find quantities like total distance, work done, or probabilities depending on the context.

To evaluate a definite integral, we perform the following steps:

• Find the antiderivative of the integrand.
• Evaluate this antiderivative at the upper and lower limits of the interval.
• Subtract the value obtained at the lower limit from the value at the upper limit.

While the process itself might seem straightforward, trigonometric substitution often makes it easier to handle complex integrals involving square roots. However, unlike indefinite integrals where we add a constant of integration \( C \), definite integrals yield a specific numerical result owing to the boundaries.

Although our original problem asks for an indefinite integral, understanding the principles behind definite integrals is crucial as it deepens comprehension of integral calculus and its applications.

Computer Algebra System (CAS)

A Computer Algebra System (CAS) is a software tool designed to perform symbolic mathematics, including algebra, calculus, and more. It's particularly useful for verifying solutions to complex integrals that might be prone to human error or challenging to solve by hand.

CAS tools such as Wolfram Alpha, MATLAB, or graphing calculators, allow students to focus more on understanding the process and less on the tedious algebraic manipulation. They can solve integrals, simplify expressions, and even provide step-by-step solutions for better learning experiences.

• For symbolic calculations, CAS can solve integrals and differentiate functions.
• For numerical approximations, CAS can produce decimal values for integrals that are difficult to express in closed form.
• CAS provides a means to check your work and explore "what-if" scenarios by adjusting functions or variables.

In evaluating the integrals like the one in the exercise, using a CAS helps verify the manual calculations and gives credibility to the taught methods. The insight gained from comparing hand-computed solutions with CAS output ensures that the concept is well understood and correctly applied.