Problem 43 - Thomas Calculus in SI Units

Question

Evaluate the integrals. $$\int \frac{d x}{\sqrt{9-x^{2}}}$$

Step-by-Step Solution

Answer

The integral evaluates to \( \arcsin\left(\frac{x}{3}\right) + C \).

Step 1: Identify the form of the integrand

The integrand \( \frac{1}{\sqrt{9-x^2}} \) resembles the derivative of the arcsine function. The derivative of \( \arcsin(x) \) is \( \frac{1}{\sqrt{1-x^2}} \). Notice that the expression under the square root is \( 9-x^2 = 3^2 - x^2 \), which suggests a trigonometric substitution might be suitable.

Step 2: Apply trigonometric substitution

Use the substitution \( x = 3 \sin(\theta) \), which implies \( dx = 3 \cos(\theta) \, d\theta \). The integral becomes \( \int \frac{3 \cos(\theta) \, d\theta}{\sqrt{9 - 9\sin^2(\theta)}} \). Using the identity \( \cos^2(\theta) = 1 - \sin^2(\theta) \), the integral simplifies to \( \int 1 \, d\theta \).

Step 3: Evaluate the simplified integral

The integral \( \int 1 \, d\theta \) is straightforward and evaluates to \( \theta + C \), where \( C \) is the constant of integration.

Step 4: Reverse the trigonometric substitution

Since \( x = 3 \sin(\theta) \), we find \( \theta = \arcsin\left(\frac{x}{3}\right) \). Hence, the solution to the original integral is \( \arcsin\left(\frac{x}{3}\right) + C \).

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a powerful technique used in calculus for simplifying integrals that involve square roots of quadratic polynomials. The idea is to transform the integral into a simpler form by using trigonometric identities.

Here's why it's helpful:

• It exploits the Pythagorean identity: For example, we know that \( heta \) in trigonometric functions often relates through \( ext{sin}^2(\theta) + ext{cos}^2(\theta) = 1 \).
• This allows the integrand to simplify nicely, often transforming the square root expression into a simple form using trigonometric functions.

In this case, by using the substitution \( x = 3 \sin(\theta) \), we closely align the integral with the derivative identity of the arcsine function. The substitution changes the variable of the integral from \( x \) to \( \theta \), making it much easier to evaluate.

The real challenge lies in choosing the correct trigonometric identity or substitution. In this example, the expression \( 9-x^2 \) triggers the use of \( ext{sin} \) because it fits the Pythagorean trigonometric identity structure.

Arcsine Function

The arcsine function, denoted as \( \arcsin(x) \), is the inverse of the sine function and is crucial in integral calculus. It helps in determining angles from given sine values.

In the context of integration, identifying an integral with a form resembling the derivative of arcsine can simplify solving it.

• Derivative of arcsine: The derivative \( \frac{d}{dx} \arcsin(x) = \frac{1}{\sqrt{1-x^2}} \) is critical for recognizing similar integral forms.
• This derivative matches the general form \( \frac{1}{\sqrt{a^2-x^2}} \), where \( a \) is a constant.

For this exercise, we rearrange the integral \( \int \frac{dx}{\sqrt{9-x^2}} \) so it can be solved using the arcsine function. By recognizing \( a = 3 \), the integral directly becomes the inverse sine function \( \theta = \arcsin\left(\frac{x}{3}\right) \). This neat transformation allows simple evaluation and mapping back to the original variable.

Calculus Integration

Integration constitutes one of the two fundamental operations of calculus, used to find areas under curves among other applications. In particular, calculus integration often involves antiderivatives and definite and indefinite integrals.

Indefinite integrals, like the one in this problem, are a primary type where we seek an antiderivative of a given function.

• Understanding the basic rules and techniques, such as substitution and integration of trigonometric identities, provides tools to evaluate complex functions.
• Integration can transform complex algebraic expressions into their respective antiderivatives, simplifying functions and solving equations.

In solving \( \int \frac{d x}{\sqrt{9-x^{2}}} \), we utilize techniques such as trigonometric substitution to find a form that can be inverted back to familiar basic functions like arcsine. Once simplified, integration yields a result that includes constants of integration. This solution is isomorphic to the problem domain, ensuring accurate representation of functions and curves.