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Chapter 4: Problem 13

In Exercises \(1-16,\) find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. \begin{equation} \text { a. } \frac{1}{2} \sec ^{2} x \quad \text { b. } \frac{2}{3} \sec ^{2} \frac{x}{3} \quad \text { c. }-\sec ^{2} \frac{3 x}{2} \end{equation}

Short Answer

Expert verified
a. \(\frac{1}{2}\tan(x)+C\); b. \(2\tan\left(\frac{x}{3}\right)+C\); c. \(-\frac{2}{3}\tan\left(\frac{3x}{2}\right)+C\).

Step by step solution

01

Identify the Basic Antiderivative

The function involves the term \( \sec^2(x) \). Recall that the antiderivative of \( \sec^2(x) \) is \( \tan(x) \). This will be the basis for solving parts a, b, and c.
02

Solve Part a

The given function is \( \frac{1}{2} \sec^2(x) \). Using the basic antiderivative, we get \( \frac{1}{2} \tan(x) + C \) where \( C \) is the constant of integration. Differentiate \( \frac{1}{2} \tan(x) \) to check: \( \frac{1}{2} \sec^2(x) \), which matches the original function.
03

Solve Part b

The given function is \( \frac{2}{3} \sec^2\left(\frac{x}{3}\right) \). Using substitution, let \( u = \frac{x}{3} \). Thus, \( \frac{du}{dx} = \frac{1}{3} \) and \( dx = 3 \, du \). The antiderivative becomes \( 2 \int \sec^2(u) \, du = 2 \tan(u) + C \) or \( 2 \tan\left(\frac{x}{3}\right) + C \). Differentiating gives \( \frac{2}{3} \sec^2\left(\frac{x}{3}\right) \), confirming our solution.
04

Solve Part c

The given function is \(-\sec^2\left(\frac{3x}{2}\right) \). Let \( u = \frac{3x}{2} \). Then \( \frac{du}{dx} = \frac{3}{2} \) and \( dx = \frac{2}{3} \, du \). The antiderivative becomes \(-\frac{2}{3} \int \sec^2(u) \, du = -\frac{2}{3} \tan(u) + C = -\frac{2}{3} \tan\left(\frac{3x}{2}\right) + C \). Check by differentiation to ensure it matches the original function.

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Key Concepts

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

Secant Function
The secant function, denoted as \( \sec(x) \), is a trigonometric function that is the reciprocal of the cosine function. This means that \( \sec(x) = \frac{1}{\cos(x)} \). It's an essential part of trigonometric calculus, often arising in various integration problems.
The derivative of \( \sec(x) \) is \( \sec(x)\tan(x) \), and the integral of \( \sec^2(x) \) is \( \tan(x) \), which makes it useful when dealing with antiderivatives. The secant function is unique, as it has vertical asymptotes where its cosine counterpart is zero, leading to complex integration scenarios.
Differentiation
Differentiation is a core concept in calculus involving the calculation of the derivative of a function. The derivative measures how a function changes as its input changes. It provides the slope of the tangent line at any point on the function.
When checking an antiderivative, differentiation plays a vital role:
  • Start by identifying the function for which you have found an antiderivative.
  • Take the derivative of your antiderivative.
  • Ensure that the result matches the original function to confirm accuracy.
For example, if you find an antiderivative to be \( \tan(x) \), differentiate it to obtain \( \sec^2(x) \), confirming the connection between differentiation and antiderivatives.
Substitution Method
The substitution method is a useful technique for solving integrals, especially when dealing with more complex expressions. It involves replacing a part of the integral with a new variable to make the problem easier to solve.
Here's how it works:
  • Select a substitution. For instance, if integrating \( \sec^2\left(\frac{x}{3}\right) \), let \( u = \frac{x}{3} \).
  • Adjust the differential, so \( \frac{du}{dx} = \frac{1}{3} \), which leads to \( dx = 3 \, du \).
  • Rewrite the integral in terms of \( u \) and \( du \), making it simpler: \( 2\int \sec^2(u) \, du \).
  • Solve the integral and substitute back the original variable to express the final antiderivative correctly.
This method is powerful, simplifying complex integrals by turning them into basic forms that are easier to tackle.
Trigonometric Integration
Trigonometric integration involves finding antiderivatives of trigonometric functions. It's crucial when solving problems involving functions like sine, cosine, and secant.
A few tips to consider:
  • Understand the basic antiderivatives of trigonometric functions. For example, \( \int \sec^2(x) \, dx = \tan(x) + C \).
  • Use trigonometric identities to transform expressions into recognizable forms.
  • Combine techniques, like substitution, to simplify integrals involving more complicated trigonometric terms.
Mastering trigonometric integration helps in solving problems beyond basic polynomials, providing analytical frameworks for more complex calculus tasks.

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Most popular questions from this chapter

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