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Chapter 5: Problem 24

Evaluate. $$ \int \frac{1}{4 \sec x} d x $$

Short Answer

Expert verified
\( \frac{1}{4} \sin x + C \)

Step by step solution

01

Simplify the Integral

The given integral is \( \int \frac{1}{4 \sec x} \, dx \). We can rewrite this as \( \int \frac{\cos x}{4} \, dx \) since \( \sec x = \frac{1}{\cos x} \). This simplifies the integral to \( \frac{1}{4} \int \cos x \, dx \).
02

Integrate the Function

The antiderivative of \( \cos x \) is \( \sin x \). Thus, we have \( \frac{1}{4} \int \cos x \, dx = \frac{1}{4} \sin x + C \), where \( C \) is the constant of integration.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Antiderivative
The term "antiderivative" is an essential concept in calculus. It refers to a function that "undoes" the operation of differentiation. In simpler terms, if a function \(F(x)\) is an antiderivative of \(f(x)\), then the derivative of \(F(x)\) is $f(x), i.e.,\(\frac{d}{dx} F(x) = f(x)\). Integrating a function essentially involves finding its antiderivative.
  • For the integral \( \int \cos x \, dx \), the antiderivative is \( \sin x \).
  • We add a constant \(C\) because indefinite integrals have infinite solutions differing by a constant.
When dealing with complex expressions, breaking them down into simpler components, like trigonometric identities, can ease finding the antiderivative. Understanding the foundational derivative rules, like how \(\sin x\) is the antiderivative of \(\cos x\), is key to mastering integration.
Trigonometric Functions
Trigonometric functions are vital in calculus, especially when dealing with periodic functions. In our problem, the \(\sec x\) function is highlighted by its trigonometric identity, \(\sec x = \frac{1}{\cos x}\). This understanding aids in simplifying and calculating integrals.
  • Converting \(\sec x\) to \(\cos x\) is a typical technique in integration to make the function more manageable.
  • Key trig identity: \(\cos x = \frac{1}{\sec x}\), makes it easier to find integrals involving secant.
Recognizing and applying these identities allows us to transform complex trigonometric integrals into simpler forms. Trigonometric integrals often require a good grasp of these identities for ease of calculation, and they frequently appear in calculus.
Definite Integral
While the current problem involves an indefinite integral, understanding definite integrals is integral (pun intended) to mastering calculus concepts. A definite integral provides the area under a curve between two specific points, written as \(\int_{a}^{b} f(x) \, dx\).
  • In contrast to indefinite integrals, which represent a family of functions, definite integrals calculate a numerical value.
  • For definite integrals, the antiderivative is evaluated from the lower limit, \(a\), to the upper limit, \(b\).
While it's not directly used in the current problem, understanding both indefinite and definite integrals allows for the full appreciation of how calculus describes physical systems and phenomena.

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Most popular questions from this chapter

Let \(f(x)=9-x^{2}\) for \(-2 \leq x \leq 3,\) and let \(P\) be the regular partition of [-2,3] into five subintervals. Find the Riemann sum \(R_{p}\) if \(f\) is evaluated at the midpoint of each subinterval of \(P\).

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