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 4.7.35
Chapter 4, Problem 4.7.35

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.


∫(−2cost) dt

Verified step by step guidance
1
Identify the integral to solve: \(\int (-2 \cos t) \, dt\).
Recall the basic integral formula for cosine: \(\int \cos t \, dt = \sin t + C\).
Apply the constant multiple rule for integrals: \(\int k f(t) \, dt = k \int f(t) \, dt\), where \(k\) is a constant.
Rewrite the integral using the constant multiple: \(\int (-2 \cos t) \, dt = -2 \int \cos t \, dt\).
Integrate \(\cos t\) and multiply by \(-2\): \(-2 \sin t + C\), where \(C\) is the constant of integration.

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 Integral

An indefinite integral represents the family of all antiderivatives of a function and is expressed with a constant of integration, C. It reverses differentiation, meaning if F'(x) = f(x), then ∫f(x) dx = F(x) + C.
Recommended video:
05:04
Introduction to Indefinite Integrals

Integration of Trigonometric Functions

Integrating trigonometric functions like cosine involves knowing their antiderivatives. For example, the integral of cos(t) dt is sin(t) + C, and constants can be factored out to simplify the process.
Recommended video:
6:04
Introduction to Trigonometric Functions

Verification by Differentiation

After finding an indefinite integral, differentiating the result should return the original integrand. This step confirms the correctness of the antiderivative and helps identify any errors in the integration process.
Recommended video:
05:53
Finding Differentials
Related Practice
Textbook Question

In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.

53. y = x * √(8 - x²)

Textbook Question

In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.

______

y = √𝓍² ― 1

Textbook Question

113. If b, c, and d are constants, for what value of b will the curve y = x^3 + bx^2 + cx + d have a

point of inflection at x = 1? Give reasons for your answer.

Textbook Question

Applications


Stopping a motorcycle The State of Illinois Cycle Rider Safety Program requires motorcycle riders to be able to brake from 30 mph (44 ft/sec) to 0 in 45 ft. What constant deceleration does it take to do that?

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.


∫(√x + ³√x) dx

Textbook Question

37. What value of a makes f(x) = x^2 +(a/x) have

a. a local minimum at x = 2?

b. a point of inflection at x = 1?