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.83
Chapter 4, Problem 4.PE.83

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.

∫ sec² s/10 ds

Verified step by step guidance
1
Identify the integral to solve: \(\int \sec^{2}\left(\frac{s}{10}\right) \, ds\).
Recall the basic integral formula: \(\int \sec^{2}(x) \, dx = \tan(x) + C\), where \(C\) is the constant of integration.
Since the argument of the secant squared function is \(\frac{s}{10}\) instead of \(s\), use substitution to handle the inner function. Let \(u = \frac{s}{10}\).
Differentiate \(u\) with respect to \(s\) to find \(du\): \(du = \frac{1}{10} ds\), which implies \(ds = 10 \, du\).
Rewrite the integral in terms of \(u\): \(\int \sec^{2}(u) \, (10 \, du) = 10 \int \sec^{2}(u) \, du\). Then integrate using the basic formula to get \(10 \tan(u) + C\). Finally, substitute back \(u = \frac{s}{10}\) to express the answer in terms of \(s\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

Key Concepts

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

Indefinite Integral

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.
Recommended video:
05:04
Introduction to Indefinite Integrals

Integration of Trigonometric Functions

Certain trigonometric functions have standard integrals, such as ∫sec²(x) dx = tan(x) + C. Recognizing these forms helps in directly integrating expressions involving trigonometric functions.
Recommended video:
6:04
Introduction to Trigonometric Functions

Substitution Method

When the integrand involves a function inside another function, substitution simplifies the integral by changing variables. For example, setting u = s/10 transforms the integral into a standard form easier to solve.
Recommended video:
07:33
Euler's Method
Related Practice
Textbook Question

Initial Value Problems

Solve the initial value problems in Exercises 89–92.


dy/dx = (𝓍 + 1/𝓍)² , y(1)= 1

Textbook Question

Applications


Liftoff from Earth A rocket lifts off the surface of Earth with a constant acceleration of 20 m/sec². How fast will the rocket be going 1 min later?

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.

∫ cos³ 𝓍/2 d𝓍

Textbook Question

54. Fermat’s principle in optics Light from a source A is reflected by a plane mirror to a receiver at point B, as shown in the accompanying figure. Show that for the light to obey Fermat’s principle, the angle of incidence must equal the angle of reflection, both measured from the line normal to the reflecting surface. (This result can also be derived without calculus. There is a purely geometric argument, which you may prefer.)

1
views
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.

__

∫ ( 3√ t + 4/t² ) dt

Textbook Question

Finding Indefinite Integrals


In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.


∫(t√t + √t) / t² dt