Problem 9 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(1-28\). $$ \int \frac{d x}{\sqrt{4 x^{2}-49}}, \quad x>\frac{7}{2} $$

Step-by-Step Solution

Answer

\( \ln \left| \frac{2x}{7} + \sqrt{\left(\frac{2x}{7}\right)^2 - 1} \right| + C \).

Step 1: Identify the integral form

The integral is of the form \( \int \frac{dx}{\sqrt{4x^2 - 49}} \). This can be recognized as resembling a standard form for trigonometric substitution, specifically involving \( \sqrt{a^2x^2 - a^2} \) which suggests the use of hyperbolic functions.

Step 2: Factor the expression inside the square root

Rewrite \( 4x^2 - 49 \) as \( 4(x^2 - \frac{49}{4}) \) to make the expression inside the square root easier to handle for substitution. This simplifies to \( 2\sqrt{x^2 - \frac{49}{4}} \).

Step 3: Choose the appropriate substitution

Since the expression now is \( \sqrt{x^2 - \left(\frac{7}{2}\right)^2} \), we can use the trigonometric substitution \( x = \frac{7}{2} \sec(\theta) \). This gives a derivative of \( dx = \frac{7}{2}\sec(\theta)\tan(\theta) \,d\theta \).

Step 4: Substitute and simplify

Substitute \( x = \frac{7}{2} \sec(\theta) \) into the integral. The expression \( \sqrt{4x^2 - 49} \) becomes \( \sqrt{\left(\frac{7}{2}\right)^2 \sec^2(\theta) - \left(\frac{7}{2}\right)^2} = \frac{7}{2}\tan(\theta) \). Thus, the integral becomes: \[ \int \frac{\frac{7}{2}\sec(\theta)\tan(\theta)\, d\theta}{\frac{7}{2}\tan(\theta)} = \int \sec(\theta) \, d\theta \].

Step 5: Integrate the simplified expression

The integral \( \int \sec(\theta) \, d\theta \) is known to be \( \ln |\sec(\theta) + \tan(\theta)| + C \), where \( C \) is the constant of integration.

Step 6: Convert back to variable x

Since \( x = \frac{7}{2} \sec(\theta) \), then \( \sec(\theta) = \frac{2x}{7} \) and \( \tan(\theta) = \sqrt{\sec^2(\theta) - 1} = \sqrt{\left(\frac{2x}{7}\right)^2 - 1} \). Substitute these back to get: \[ \ln \left| \frac{2x}{7} + \sqrt{\left(\frac{2x}{7}\right)^2 - 1} \right| + C \] as the final expression in terms of \( x \).

Key Concepts

Hyperbolic Functions

Hyperbolic functions, such as the hyperbolic sine (\(\sinh\)) and hyperbolic cosine (\(\cosh\)), are analogues of the trigonometric functions but for hyperbolas rather than circles. These functions often arise in problems of calculus due to their properties that mimic certain trigonometric identities, making them suitable for particular types of integrations, especially those involving quadratic expressions.
A key characteristic of hyperbolic functions is their definition through exponential functions:

• \(\sinh(x) = \frac{e^x - e^{-x}}{2}\)
• \(\cosh(x) = \frac{e^x + e^{-x}}{2}\)

Hyperbolic functions can simplify integrals involving expressions like \(\sqrt{x^2 + a^2}\) due to their relationships similar to Pythagorean identities in trigonometry.
Using hyperbolic identities such as \(\cosh^2(x) - \sinh^2(x) = 1\) can be a powerful method in evaluations where traditional trigonometric substitutions are cumbersome.

Integral Calculus

Integral calculus is the branch of mathematics concerned with finding the total or accumulation of quantities. It often deals with the calculation of areas under curves or the total size accumulation when given rates of change.
At the core of integral calculus is the concept of an "integral," which can be either definite or indefinite:

• Indefinite integrals, denoted \(\int f(x) \, dx\), represent a family of functions whose derivative is \(f(x)\).
• Definite integrals, written as \(\int_a^b f(x) \, dx\), calculate the net area under \(f(x)\) from \(a\) to \(b\).

Trigonometric substitution is a technique used in integral calculus where a trigonometric function is introduced to simplify the integration process. This method is particularly helpful when dealing with integrals involving roots of quadratic expressions, transforming the problem into one of standard trigonometric integrals.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variable where both sides of the identity are defined. Basic identities stem from the definitions of sine, cosine, and tangent, and include:

• \(\sin^2(x) + \cos^2(x) = 1\)
• \(1 + \tan^2(x) = \sec^2(x)\)
• \(1 + \cot^2(x) = \csc^2(x)\)

These identities are pivotal in manipulating expressions and integrals in calculus.
In trigonometric substitution, one often uses identities to convert square roots into trigonometric functions that are easier to integrate. For example, \(\sec^2(\theta) - 1 = \tan^2(\theta)\) is used in the process of resolving complications within expressions involving square roots.
Understanding these identities and how to apply them facilitates problem-solving in calculus, especially when traditional methods of integration fall short.