Skip to main content
Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.PE.86
Chapter 4, Problem 4.PE.86

Finding Indefinite Integrals
Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ sec θ/3 tan θ/3 dθ

Verified step by step guidance
1
Recognize the integral to solve: \(\int \sec\left(\frac{\theta}{3}\right) \tan\left(\frac{\theta}{3}\right) \, d\theta\).
Recall the derivative identity: \(\frac{d}{dx} \sec x = \sec x \tan x\). This suggests that the integral of \(\sec x \tan x\) with respect to \(x\) is \(\sec x + C\).
Use a substitution to handle the argument \(\frac{\theta}{3}\). Let \(u = \frac{\theta}{3}\), so that \(d\theta = 3 \, du\).
Rewrite the integral in terms of \(u\): \(\int \sec u \tan u \cdot 3 \, du = 3 \int \sec u \tan u \, du\).
Integrate with respect to \(u\) using the known formula: \(\int \sec u \tan u \, du = \sec u + C\). Then substitute back \(u = \frac{\theta}{3}\) to express the answer in terms of \(\theta\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Indefinite Integrals

An indefinite integral represents the most general antiderivative of a function, expressed with a constant of integration (C). It reverses differentiation and provides a family of functions whose derivative is the integrand. Understanding indefinite integrals is essential for solving problems without specified limits.
Recommended video:
05:04
Introduction to Indefinite Integrals

Integration Techniques for Trigonometric Functions

Integrating trigonometric functions often requires recognizing standard forms or using substitution. For example, integrals involving secant and tangent functions can be simplified by recalling derivatives like d/dx(sec x) = sec x tan x. Familiarity with these relationships helps in guessing and verifying antiderivatives.
Recommended video:
6:04
Introduction to Trigonometric Functions

Verification by Differentiation

After finding an antiderivative, differentiating it should return the original integrand. This step confirms the correctness of the solution. Checking answers by differentiation is a crucial practice to ensure no mistakes were made during integration.
Recommended video:
05:53
Finding Differentials
Related Practice
Textbook Question

Initial Value Problems

Solve the initial value problems in Exercises 89–92.

d^3 r/dt^3 = - cos t; r''(0) = r'(0) = 0 , r(0) = -1

Textbook Question

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ 1/( r + 5)²dr

Textbook Question

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ (𝓍³ + 5𝓍 ―7) d𝓍

Textbook Question

Identifying Extrema


In Exercises 53–60:


a. Find the local extrema of each function on the given interval, and say where they occur.


f(x) = sin x − cos x,0 ≤ x ≤ 2π

Textbook Question

Identifying Extrema


In Exercises 41–52:


a. Identify the function’s local extreme values in the given domain, and say where they occur.


f(x) = (x + 1)², −∞ < x ≤ 0

Textbook Question

Identifying Extrema


In Exercises 19–40:


a. Find the open intervals on which the function is increasing and those on which it is decreasing.


f(r) = 3r³ + 16r