Problem 9 - Calculus: Early Transcendental Functions

Question

Use Part I of the Fundamental Theorem to compute each integral exactly. $$\int_{0}^{\pi / 4} \sec x \tan x d x$$

Step-by-Step Solution

Answer

The definite integral of \(\sec(x) \tan(x)\) from 0 to \(\pi/4\) equals \(\sqrt{2} - 1\).

Step 1: Identify the Antiderivative

Identify the antiderivative of the expression inside the given integral. The antiderivative of \(\sec(x) \tan(x)\) is \(\sec(x)\) since the derivative of \(\sec(x)\) is \(\sec(x) \tan(x)\).

Step 2: Apply Fundamental Theorem of Calculus

Apply the Fundamental Theorem of Calculus, which states that the definite integral of a function f from a to b is F(b) - F(a), where F is an antiderivative of f. Here, the antiderivative F(x) is \(\sec(x)\), a = 0, and b = \(\pi/4\). Hence, the result of the integral is \(\sec(\pi/4) - \sec(0)\).

Step 3: Calculate

Calculate the result of the definite integral by plugging in the values for a and b into the antiderivative. \(\sec(\pi/4) - \sec(0)\) equals \(\sqrt{2} - 1\).

Key Concepts

Definite Integrals

Definite integrals play a vital role in calculus, especially when determining the total accumulation of a quantity over a certain interval. Unlike indefinite integrals, definite integrals provide a specific numerical value. This is achieved by considering the bounds of integration, denoted by the limits at the bottom and top of the integral symbol. In mathematical terms, a definite integral from a to b of a function f(x) is represented as: \[\int_{a}^{b} f(x) \, dx\]

• The expression inside the integral describes the function f(x) being integrated.
• The limits, a and b, define the interval over which the function is evaluated.

The definite integral can be interpreted as the signed area under the curve of f(x) from x = a to x = b. This area can be calculated precisely using the antiderivative of the function, applying the Fundamental Theorem of Calculus. Hence, definite integrals are essential in many areas, from physics to economics, where measuring total change is necessary.

Antiderivative

An antiderivative is a function whose derivative is the original function that you started with. It is essentially the reverse process of differentiation. If you have a function f(x), an antiderivative F(x) satisfies the condition:\[F'(x) = f(x)\]Finding an antiderivative is akin to "undoing" the derivative. For example, the derivative of \( \sec(x) \) is \( \sec(x) \tan(x) \), thus making \( \sec(x) \) an antiderivative of \( \sec(x) \tan(x) \).

• Antiderivatives are crucial for evaluating definite integrals.
• They help in reversing the differentiation process, allowing for integration.

Knowing the antiderivative directly impacts the computation of definite integrals, a concept deeply ingrained in the Fundamental Theorem of Calculus. This theorem allows for using antiderivatives to determine the exact value of definite integrals. Consequently, mastering antiderivatives equips you with the tools to tackle a broad array of integral problems.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are essential in describing periodic phenomena. They often appear in calculus problems related to integration and differentiation. In this particular exercise, secant (sec) and tangent (tan) functions are used. Understanding trigonometric identities and derivatives is key in solving such integrals.

• The derivative of \( \sec(x) \) is: \( \sec(x) \tan(x) \).
• Secant (\( \sec(x) \)) is the reciprocal of cosine (\( \cos(x) \)).
• Trigonometric functions often relate through their derivatives and antiderivatives.

In calculus, these functions' properties, such as periodicity and symmetry, can simplify the process of finding antiderivatives. Knowing these functions and their related identities allows you to simplify the integrals before applying the Fundamental Theorem of Calculus for evaluating them, thereby making trigonometry a crucial element in calculus solutions.

Step-by-Step Solution

Solving integration problems using a step-by-step approach ensures clarity and accuracy in arriving at the correct answer. Here’s a breakdown of using Part I of the Fundamental Theorem of Calculus, demonstrated through the exercise:1. **Identify the Antiderivative:** Start by pinpointing the antiderivative of the function in the integral. For \( \sec(x) \tan(x) \), the antiderivative is \( \sec(x) \).
2. **Apply the Fundamental Theorem of Calculus:** Use this theorem, which states: \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \), where F is the antiderivative of f. This step translates the function into a usable formula.
3. **Calculate the Integral:** Substitute the limits of integration (0 and \( \pi/4 \) in this exercise) into the antiderivative, giving \( \sec(\pi/4) - \sec(0) \). By computing this, the definite integral evaluates to \( \sqrt{2} - 1 \).
This structured method ensures you follow all necessary procedures to compute integrals accurately, providing an excellent foundation for solving more complex problems. By dissecting each phase in the solution, you reinforce the understanding of calculus concepts involved.