Problem 41 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Evaluate the integrals. Not all require a trigonometric substitution. Choose the simplest method of integration. \(\int_{\frac{2}{\sqrt{3}}}^{2} \frac{\sqrt{x^{2}-1}}{x} d x\)

Step-by-Step Solution

Answer

The integral diverges.

Step 1: Trig Substitution

First, make a trig substitution to simplify the integrand. Let \(x = \sec(\theta)\). Then, \(dx = \sec(\theta)tan(\theta)d\theta\), \(\sqrt{x^{2}-1} = |\tan(\theta)|\), which is just \(\tan(\theta)\) for \(\theta \in [0,\frac{\pi}{2}]\). The limits of integration will also change. When \(x=\frac{2}{\sqrt{3}}\), \(\theta=\frac{\pi}{3}\), and when \(x=2\), \(\theta=\frac{\pi}{2}\).

Step 2: Substitute in the Integrals

The given integral becomes: \(\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \tan(\theta) d\theta\).

Step 3: Find the Anti-Derivative

Find the anti-derivative of \(\tan(\theta)\). The integral of \(\tan(\theta)\) with respect to \(\theta\) is \(-ln|cos(\theta)|\). So the new expression to evaluate is: \(-ln|cos(\frac{\pi}{2})| + ln|cos(\frac{\pi}{3})|\).

Step 4: Evaluate the Definite Integral

Now substitute the new bounds of integration. This results in: \(\ln 2 - ln 0 \). Logarithm of zero is undefined. Therefore the integral diverges.

Key Concepts

Trigonometric Substitution

When you're faced with integrals involving square roots like \( \sqrt{x^2 - 1} \), trigonometric substitution becomes a handy tool. This method exploits the identities of trigonometric functions to simplify the integrand. In this case, the substitution \( x = \sec(\theta) \) was chosen because it turns the expression \( \sqrt{x^2 - 1} \) into \( \tan(\theta) \). This works due to the trigonometric identity \( \sec^2(\theta) - 1 = \tan^2(\theta) \).
Using trigonometric substitutions involves:

• Identifying the right trigonometric identity that matches your integrand's structure.
• Making the right substitutions for \( x \) and \( dx \).
• Updating the limits of integration to the new variable \( \theta \).

These steps simplify the integral, making it easier to solve.

Definite Integrals

A definite integral, unlike an indefinite integral, evaluates the function over a specific interval. When dealing with definite integrals, you're not just finding an anti-derivative, but also answering the question "How much area is there under this curve between these two points?"
In the original problem, the definite integral \( \int_{\frac{2}{\sqrt{3}}}^{2} \frac{\sqrt{x^{2}-1}}{x} \, dx \) required updating the limits during the trigonometric substitution process. This transformation is crucial, as it maintains the integral’s boundary definitions as the variable changes. After substitution, the integral becomes \( \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \tan(\theta) \, d\theta \).
Evaluating definite integrals involves substituting these evaluated bounds back into the anti-derivative function to determine the net area, considering both the upper and lower limits.

Anti-Derivatives

Finding an anti-derivative is essentially computing the reverse process of differentiation. This notion is integral (pun intended!) in solving integrals.
In the given exercise, the anti-derivative of \( \tan(\theta) \) is calculated. The integral of \( \tan(\theta) \) is \(-\ln|\cos(\theta)|\). This result comes from understanding how logarithmic identities relate to trigonometric functions.
The process includes:

• Recognizing the function to be integrated.
• Knowing the corresponding anti-derivative (e.g., having integral formulas handy).
• Applying integration techniques like trigonometric substitution to simplify the function into a more recognizable form.

Anti-derivatives bridge the gap between the derivative of a function and the area under its curve.

Divergent Integrals

Integrals can sometimes lead to expressions that don't converge to a finite value. These are known as divergent integrals. In the context of definite integrals, this divergence arises when substitutions or function evaluations yield undefined outcomes.
In this exercise, evaluating the integral resulted in \( \ln|\cos(\frac{\pi}{2})| \), which becomes \( \ln(0) \). The value \( \ln(0) \) is undefined because the logarithm of zero doesn't exist in the realm of real numbers.
To determine if an integral is divergent:

• Carefully substitute the bounds after calculating the anti-derivative.
• Check for any expressions or values that might not be well-defined (e.g., division by zero or logarithms of non-positive values).

Recognizing divergent integrals is crucial, as it signals that the system or area being measured does not have a bounded finite value.