Problem 11 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(1-26\) $$ \int_{0}^{\pi / 3} 2 \sec ^{2} x d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( 2\sqrt{3} \).

Step 1: Identify the Integral

The integral we need to evaluate is \( \int_{0}^{\pi / 3} 2 \sec^{2} x \, dx \). This is a definite integral over the interval \([0, \pi/3]\).

Step 2: Use a Known Antiderivative

Recall that the derivative of \( \tan x \) is \( \sec^2 x \). Hence, the antiderivative of \( \sec^2 x \) is \( \tan x \). The integral becomes \( 2 \int_{0}^{\pi / 3} \sec^{2} x \, dx = 2 [ \tan x ]_{0}^{\pi / 3} \).

Step 3: Evaluate the Antiderivative

First, find \( \tan(\pi/3) \) which is \( \sqrt{3} \), and \( \tan(0) \) which is \( 0 \). Substitute these into the antiderivative: \( [ \tan(\pi/3) ] - [ \tan(0) ] = \sqrt{3} - 0 = \sqrt{3} \).

Step 4: Multiply by the Constant

Since the integral involves a constant multiple of 2, multiply the result of the antiderivative evaluation by 2: \( 2 \times \sqrt{3} = 2\sqrt{3} \).

Step 5: Final Answer

The value of the definite integral \( \int_{0}^{\pi / 3} 2 \sec^{2} x \, dx \) is \( 2\sqrt{3} \).

Key Concepts

Antiderivative

The concept of an antiderivative is fundamental in calculus. Simply put, an antiderivative, also known as the indefinite integral, is a function whose derivative yields the original function. For instance, if you have a function like \( f(x) = \sec^2 x \), finding its antiderivative means figuring out which function, when differentiated, gives you \( f(x) \). In this specific exercise, it's important to recognize that the antiderivative of \( \sec^2 x \) is \( \tan x \). Knowing this allows us to compute the definite integral effectively.

When solving integration problems, identifying known antiderivatives can significantly simplify the process. Remember, the power of integration lies in reversing differentiation, so recognizing these crucial relationships is key.

Trigonometric Functions

Trigonometric functions like \( \sec x \), \( \tan x \), \( \sin x \), and \( \cos x \) are essential tools in calculus. They help model periodic phenomena and are widely used in numerous applications. In the exercise you're working on, dealing with \( \sec^2 x \) is important.

• The secant function, denoted by \( \sec x \), is the reciprocal of the cosine function: \( \sec x = \frac{1}{\cos x} \).
• \( \tan x \), the tangent function, is the ratio of sine to cosine: \( \tan x = \frac{\sin x}{\cos x} \).

In the realm of integration, knowing that the derivative of \( \tan x \) is \( \sec^2 x \) is invaluable. It allows for seamless transition from recognizing and evaluating the integral \( \int \sec^2 x \, dx \) to swiftly calculating it using its antiderivative. These trigonometric identities and properties are not just theoretical—they dramatically simplify practical calculations in calculus.

Calculus Exercise

Tackling calculus exercises effectively requires breaking them down into manageable steps. Understanding the problem you're solving is fundamental to finding the solution. In this specific exercise, initially, you have to identify it as a definite integral: \( \int_{0}^{\pi/3} 2 \sec^2 x \, dx \).

• Start by determining the antiderivative, which we found to be \( \tan x \) for \( \sec^2 x \).
• Evaluate the antiderivative at the given limits, substituting values like \( \tan(\pi/3) = \sqrt{3} \) and \( \tan(0) = 0 \).
• Compute the expression \( 2 \left[ \tan x \right]_{0}^{\pi/3} \), leading you to the result \( 2 \times \sqrt{3} \).

This systematic approach not only makes the exercise easier but also enhances your understanding of how integration is applicable. Such exercises don't just improve your math skills; they develop your ability to think logically and solve complex problems step by step.