Problem 18 - Calculus: Early Transcendental Functions

Question

Use Part I of the Fundamental Theorem to compute each integral exactly. $$\int_{0}^{\pi / 3} \frac{3}{\cos ^{2} x} d x$$

Step-by-Step Solution

Answer

The exact value of the integral is \(3\sqrt{3}\).

Step 1: Recognize the integrand function

First, recognize the integrand in its standard form. In this case, the integrand is the function \(f(x) = \frac{3}{\cos ^{2} x}\), which is the same as \(3 \sec^2 x\). Therefore, the integral could be rewritten as \( \int_{0}^{\pi / 3} 3 \sec^{2} x d x\).

Step 2: Find the antiderivative

The next step is to find the antiderivative of the integrand. The antiderivative of \(3 \sec^2 x\) is \(3 \tan x\).

Step 3: Apply the Fundamental Theorem of Calculus

Now, use the Fundamental Theorem of Calculus Part I to compute the integral. This theorem states that \( \int_{a}^{b} f(x) dx = F(b) - F(a)\), where \( F(x)\) is the antiderivative of \( f(x)\). In this case, \( F(x) = 3 \tan x\). Therefore, the integral is \( F(\pi / 3) - F(0)\) , which simplifies to \( 3 \tan (\pi / 3) - 3 \tan (0)\).

Step 4: Compute the integral

Finally, compute the value of the integral by plugging in the values into the equation obtained from the previous step: \( 3 \tan (\pi / 3) - 3 \tan (0) = 3\sqrt{3} - 0 = 3\sqrt{3}\).

Key Concepts

Definite Integral

A definite integral, such as \( \int_{a}^{b} f(x) \, dx \), calculates the area under a curve from point \(a\) to point \(b\) on the x-axis. It is one of the key concepts in calculus and ties directly into real-life applications, like determining the total distance traveled over a period. This form of integration provides us with a numerical value that represents this area.

• The integral sign \( \int \) signifies integration.
• \( f(x) \) is the integrand, the function you are integrating.
• \( dx \) indicates the variable of integration.
• The limits of integration \(a\) and \(b\) mark the interval on the x-axis.

When computing a definite integral, we use the Fundamental Theorem of Calculus, which connects differentiation and integration, thereby allowing us to find the exact value.

Antiderivative

An antiderivative is essentially the reverse of a derivative. It is a function \( F(x) \) whose derivative is the original function \( f(x) \). Antiderivatives are crucial in the process of integration as they enable us to evaluate definite integrals through the Fundamental Theorem of Calculus.
In the exercise provided, the function \( f(x) = 3 \sec^2 x \) has an antiderivative \( F(x) = 3 \tan x \).
When solving for the integral, finding this antiderivative allows you to compute the definite integral value. This requires an understanding of how functions transform through differentiation, effectively undoing it via integration.

Trigonometric Integration

Trigonometric integration involves integrating functions that include trigonometric expressions, such as \( \sin x, \cos x, \tan x, \) and their reciprocals. These functions are pervasive in calculus due to their many properties and relationships.

• It often requires knowledge of trigonometric identities to simplify and solve integrals.
• Understanding the derivatives and antiderivatives of trigonometric functions is essential.

In this context, the integral \( \int 3 \sec^2 x \, dx \) becomes manageable because the antiderivative of \( \sec^2 x \) is a recognized result: \( \tan x \). This familiarity allows for rapid computation and is central to trigonometric integration.

Secant Function

The secant function, \( \sec x \), is one of the main trigonometric functions and is defined as the reciprocal of the cosine function, i.e., \( \sec x = \frac{1}{\cos x} \). It frequently appears in calculus problems, especially those involving integration and differentiation.
When you encounter expressions like \( \frac{3}{\cos^2 x} \), recognizing it as \( 3 \sec^2 x \) can simplify the integration process significantly.

• Secant, alongside other trigonometric functions, forms a basis for solving complex integrals.
• It simplifies integration by relying on its derivatives and antiderivatives known from basic calculus principles.

By understanding the properties and transformations of the secant function, you can more easily tackle trigonometric integrals like the one in the exercise above.