Problem 71 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(71-94\) $$ \int \frac{d x}{\sqrt{9-x^{2}}} $$

Step-by-Step Solution

Answer

\( \int \frac{d x}{\sqrt{9-x^{2}}} = \arcsin\left(\frac{x}{3}\right) + C \)

Step 1: Understand the problem

The given integral is \( \int \frac{d x}{\sqrt{9-x^{2}}} \). This integral resembles a standard trigonometric integral formula, where the integrand contains a square root of a difference of squares.

Step 2: Apply the trigonometric substitution

Since the integrand has the form \( \sqrt{9 - x^2} \), we can use the substitution \( x = 3 \sin \theta \). This substitution implies \( dx = 3 \cos \theta \, d\theta \) and \( \sqrt{9 - x^2} = \sqrt{9 - 9 \sin^2 \theta} = 3 \cos \theta \).

Step 3: Rewrite the integral

Substitute \( x = 3 \sin \theta \) and \( dx = 3 \cos \theta \, d\theta \) into the integral: \[ \int \frac{dx}{\sqrt{9-x^{2}}} = \int \frac{3 \cos \theta \, d\theta}{3 \cos \theta} = \int d\theta \]

Step 4: Evaluate the integral

The integral of \( 1 \) with respect to \( \theta \) is simply \( \theta + C \), where \( C \) is the constant of integration.

Step 5: Back-substitute to get x in terms of \( \theta \)

Since \( x = 3 \sin \theta \), we have \( \theta = \arcsin\left(\frac{x}{3}\right) \). Thus, the integral evaluates to \( \arcsin\left(\frac{x}{3}\right) + C \).

Key Concepts

Integrals

Integrals are a fundamental concept in calculus, representing the accumulation of quantities and the area under curves. When you see a symbol like \( \int \), it signifies an integral that involves summing up infinitesimally small parts over a certain range. In the given exercise, we have an integral that needs evaluation: \[ \int \frac{d x}{\sqrt{9-x^{2}}} \]Let's break down how we tackle such problems. The main goal is to find a function whose derivative matches the given function inside the integral (known as the integrand). Generally, integrals can be classified into definite and indefinite integrals:

• Definite integrals have bounds and compute the net area under the curve over a specific interval.
• Indefinite integrals, like the one in the task, do not have specified limits and include a constant of integration \( C \) since they represent a family of functions.

Integrals are foundational in mathematics, playing crucial roles in physics, engineering, and beyond. They help model a variety of real-world quantities, from distances and areas to accumulations of quantities like charge and mass.

Trigonometric Integrals

Trigonometric integrals involve integrating functions containing trigonometric functions like sine, cosine, and more. In many calculus problems, especially involving roots and squares inside the integrals, trigonometric identities and substitutions simplify the integration process. In our exercise, we encounter an integrand that includes \( \sqrt{9 - x^2} \), which is reminiscent of the Pythagorean identity \( \sin^2\theta + \cos^2\theta = 1 \). By using the trigonometric substitution \( x = 3 \sin\theta \), the integral transforms to involve purely trigonometric terms. Here's why this is beneficial:

• The substitution utilizes the fact that for a right triangle, if \( x = 3 \sin\theta \), then \( \sqrt{9 - x^2} = 3 \cos\theta \).
• This turns the original integral into \( \int d\theta \), which is much easier to integrate.

These types of integrals often require you to carefully consider the relationships between the functions involved to simplify the problem into a more recognizable form. Mastering trigonometric integrals opens up the ability to solve more complex mathematical and practical problems.

Integration Techniques

Integration techniques are strategies used to evaluate integrals that don't immediately appear in a basic form. Some integrals, such as our example, need advanced strategies like trigonometric substitutions. Key techniques include:

• Substitution: Simplifies the integral by changing the variable, often to switch a complicated function with a simpler trigonometric or exponential counterpart.
• Integration by Parts: Useful for products of functions, based on the product rule for differentiation.
• Partial Fractions: Breaks down complex rational functions into simpler fractions that can be integrated separately.

The trigonometric substitution used in our example strategically paralleled the expression \( \sqrt{9 - x^2} \) with the identity \( \cos^2\theta = 1 - \sin^2\theta \). After substitution, the integral \( \int \frac{3 \cos\theta \ d\theta}{3 \cos\theta} \) converted into \( \int d\theta \), simplifying the evaluation process significantly. Mastering these techniques ensures you can tackle a wide variety of integration problems with ease.