Question

In the following exercises, evaluate each integral in terms of an inverse trigonometric function. \(\int_{1}^{2 / \sqrt{3}} \frac{d x}{|x| \sqrt{x^{2}-1}}\)

Short answer

The integral evaluates to \( \frac{\pi}{6} \).

Step-by-step solution

Recognize the Integral Form

This integral can be rewritten to resemble a standard form for an inverse trigonometric function. The integral \( \int \frac{1}{|x| \sqrt{x^2 - a^2}} \, dx \) matches the derivative of the inverse hyperbolic cosine function, \( \text{arcsec}(x) \), when \( a = 1 \).

Identify Boundaries

We need to evaluate the definite integral from \( x = 1 \) to \( x = \frac{2}{\sqrt{3}} \). In this case, the formula \( \int \frac{1}{|x| \sqrt{x^{2} - 1}} \ dx = \text{arcsec}|x| + C \) can be used, with no need to put the absolute value for positive x.

Apply the Inverse Function

Evaluate the expression \( \text{arcsec}(x) \) at the boundaries, so we calculate \( \text{arcsec}(\frac{2}{\sqrt{3}}) - \text{arcsec}(1) \).

Evaluate \(\text{arcsec}(x)\) at the Bounds

- \( \text{arcsec}(1) \) equals \( 0 \) because \( \text{arcsec}(x) \) yields \( 0 \) when \( x = 1 \).- For \( \text{arcsec}(\frac{2}{\sqrt{3}}) \), use the identity \( \text{arcsec}(x) = \cos^{-1}(\frac{1}{x}) \). Hence \( \text{arcsec}(\frac{2}{\sqrt{3}}) = \cos^{-1}(\frac{\sqrt{3}}{2}) = \frac{\pi}{6} \).

Compute the Result

Now substitute back: \( \frac{\pi}{6} - 0 = \frac{\pi}{6} \). Thus, the definite integral evaluates to \( \frac{\pi}{6} \).

Key concepts

Definite Integrals

Definite integrals are a fundamental concept in calculus. They are used to calculate the total accumulation of a quantity, such as area under a curve, between two points. Unlike indefinite integrals, which represent a family of functions, definite integrals give a specific number, often referred to as the "net area" between the curve and the x-axis over an interval.

In solving definite integrals, we work within set boundaries. This means evaluating the function at the upper boundary, then subtracting the result of the evaluation at the lower boundary. The notation \[ \int_{a}^{b} f(x) \, dx \] stands for the definite integral from \( x = a \) to \( x = b \) of the function \( f(x) \). Here, \( a \) and \( b \) are known as the limits of integration.

If we consider a problem involving definite integrals, it's crucial to ensure the function is continuous within the limits. Calculating a definite integral involves:

• Recognizing the form of the integral
• Identifying an appropriate antiderivative
• Applying the Fundamental Theorem of Calculus

This theorem allows us to find the value of a definite integral by subtracting the value of the antiderivative at the lower limit from the value at the upper limit.

Arcsecant Function

The arcsecant function, denoted as \( \text{arcsec}(x) \), is the inverse of the secant function. This inverse trigonometric function helps evaluate integrals like the one in the exercise. Understanding arcsecant is key when the integral involves expressions like \( \frac{1}{|x| \sqrt{x^2 - a^2}} \), which relates to inverse trigonometric functions.

The arcsecant of a number \( x \) corresponds to the angle \( \theta \) where \( \sec(\theta) = x \) and \( \theta \) lies in the range \( \left[0, \frac{\pi}{2}\right) \cup \left(\frac{\pi}{2}, \pi\right] \). Essentially,\[ \text{arcsec}(x) = \cos^{-1}\left(\frac{1}{x}\right) \]

Computing \( \text{arcsec} \) requires understanding its relationship with the cosine function. If required, transformations utilizing this inverse function can simplify evaluating definite integrals. For example, in this situation:

• \( \text{arcsec}(1) = 0 \) because \( \sec(0) = 1 \)
• For \( \text{arcsec}(\frac{2}{\sqrt{3}}) \), use \( \cos^{-1}(\frac{\sqrt{3}}{2}) = \frac{\pi}{6} \)

The arcsecant function is an essential tool for solving specific integrals that include inverse trigonometric forms.

Integral Evaluation

When tasked with evaluating integrals, especially those of inverse trigonometric functions, we follow a systematic approach. This helps simplify and solve the given integral accurately. Let's explore the steps involved in evaluating such integrals.

First, recognize the integral's structure by matching it to known forms involving inverse trigonometric functions. These often mask as derivatives of functions like arcsec.

• Check the integrand's resemblance to a base form
• Identify which inverse function matches the structure

After identifying the inverse function involved, apply known identities to express the integral in more manageable terms. By recognizing these forms, we can easily transition from the integral into its evaluated outcomes.

Next, apply the definite integral limits, which typically involve substituting bounds into the inverse function. Make sure to replace the unknown \( x \) values with the limits of integration. In this context, the function values are subtracted based on the boundary application:

• Compute \( \text{arcsec}(\text{upper bound}) - \text{arcsec}(\text{lower bound}) \)

Evaluating the integral in the original problem thus requires recognizing the pattern, applying the function involved, and calculating the boundary subtraction, thus simplifying the solution to a definite answer.