Problem 18 - University Calculus: Early Transcendentals

Question

Find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. a. \(\sec x \tan x\) b. \(4 \sec 3 x \tan 3 x\) c. \(\sec \frac{\pi x}{2} \tan \frac{\pi x}{2}\)

Step-by-Step Solution

Answer

a. \(\sec x + C\); b. \(\frac{4}{3} \sec 3x + C\); c. \(\frac{2}{\pi} \sec \frac{\pi x}{2} + C\).

Step 1: Understanding Antiderivatives

An antiderivative of a function is a function whose derivative is the original function. In this exercise, we are finding an antiderivative of each given function. We'll identify the basic antiderivatives and use properties of integration.

Step 2: Finding an Antiderivative for \( \sec x \tan x \)

Recognizing that the derivative of \( \sec x \) is \( \sec x \tan x \), the antiderivative of \( \sec x \tan x \) is \( \sec x + C \), where \( C \) is the constant of integration.

Step 3: Check by Differentiation for \( \sec x \tan x \)

Differentiate \( \sec x + C \). The derivative of \( \sec x \) is \( \sec x \tan x \). Hence, \( \frac{d}{dx}(\sec x + C) = \sec x \tan x \), which confirms our antiderivative is correct.

Step 4: Finding an Antiderivative for \( 4 \sec 3x \tan 3x \)

Recognize that the derivative of \( \sec 3x \) is \( 3\sec 3x \tan 3x \). The antiderivative of \( 4\sec 3x \tan 3x \) is \( \frac{4}{3} \sec 3x + C \).

Step 5: Check by Differentiation for \( 4 \sec 3x \tan 3x \)

Differentiate \( \frac{4}{3} \sec 3x + C \). Using the chain rule, the derivative is \( 4\sec 3x \tan 3x \). This confirms our antiderivative \( \frac{4}{3} \sec 3x + C \) is correct.

Step 6: Finding an Antiderivative for \( \sec \frac{\pi x}{2} \tan \frac{\pi x}{2} \)

Recognize that the derivative of \( \sec \frac{\pi x}{2} \) is \( \frac{\pi}{2} \sec \frac{\pi x}{2} \tan \frac{\pi x}{2} \). The antiderivative is thus \( \frac{2}{\pi} \sec \frac{\pi x}{2} + C \).

Step 7: Check by Differentiation for \( \sec \frac{\pi x}{2} \tan \frac{\pi x}{2} \)

Differentiate \( \frac{2}{\pi} \sec \frac{\pi x}{2} + C \). Again, using the chain rule, the derivative is \( \sec \frac{\pi x}{2} \tan \frac{\pi x}{2} \). This confirms our antiderivative \( \frac{2}{\pi} \sec \frac{\pi x}{2} + C \) is correct.

Key Concepts

Integration

Integration is a fundamental concept in calculus, and it's often referred to as the reverse of differentiation. It involves finding a function whose derivative is given, which is known as finding an antiderivative. The primary goal of integration is to determine the area under a curve or to reverse the process of differentiation to find the original function from its derivative. It's particularly useful in physics, engineering, and economics.

Some key points about integration include:

• Indefinite integral: This represents a family of functions whose derivative is the original function. The notation for this is \( \int f(x) \, dx \), which reads "integral of \( f(x) \) with respect to \( x \)."
• Definite integral: This involves integrating within specified limits, providing the area under the curve from one point to another.
• Antiderivative: Also known as the indefinite integral, it involves finding a function whose derivative is the given function.
• Constant of integration \( C \): When finding an indefinite integral, you must always add \( C \), representing an infinite number of vertical shifts.

Integrating functions requires familiarity with basic integration rules and techniques. For trigonometric functions, recognizing derivatives of known functions like \( \sec x \tan x \) is crucial. Here, the exercise focused on identifying the antiderivative by reversing known derivatives.

Differentiation

Differentiation is the process of finding the derivative of a function. The derivative represents the rate of change of a function with respect to a variable. When you differentiate, you are essentially determining how a function changes at each point. Derivatives are used to find slopes of curves, optimize functions, and solve problems involving motion and rates.

Important aspects of differentiation include:

• Basic derivative rules: Familiarity with rules like the power, product, quotient, and chain rules helps simplify the process of differentiation.
• Chain rule: This is used when differentiating composite functions, crucial for functions involving products of trigonometric functions like \( \sec x \tan x \).
• Check by differentiation: When finding an antiderivative, checking your work involves taking the derivative of your solution to see if you arrive back at the original function.

For instance, in the original exercise, once the antiderivatives like \( \sec x \) or \( \rac{4}{3} \sec 3x \) were found, checking them by differentiation was essential. Always remember, the derivative of a correctly found antiderivative should bring you right back to the original function.

Trigonometric Functions

Trigonometric functions extend beyond their geometric origins and are crucial in calculus, particularly for finding derivatives and antiderivatives. They include sine, cosine, tangent, secant, cosecant, and cotangent. These functions are periodic and play a key role in modeling cyclical phenomena such as sound waves, light waves, and mechanical vibrations.

Some essential points about trigonometric functions are:

• Common derivatives: Knowing the derivatives of basic trigonometric functions (e.g., derivative of \( \sin x \) is \( \cos x \); derivative of \( \tan x \) is \( \sec^2 x \)) helps in finding antiderivatives found in exercises.
• Common antiderivatives: Recognizing that the antiderivative of a derivative involves reversing these processes, such as realizing \( \sec x \tan x \) is the derivative of \( \sec x \).
• Chain rule with trigonometric functions: Often requires using the chain rule to accommodate within more complex argument expressions like \( \sec 3x \) or \( \sec \rac{\pi x}{2} \).

In this specific exercise, recognizing the derivatives of trigonometric functions was crucial. For example, knowing that \( \frac{d}{dx}(\rac{2}{\pi} \sec \rac{\pi x}{2}) \) must return \( \sec \rac{\pi x}{2} \tan \rac{\pi x}{2} \) ensured the accuracy of the antiderivative.