Skip to main content
Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 4 - Applications of DerivativesProblema 4.7.47
Capitolo 4, Problema 4.7.47

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.


∫(1 + cos 4t)/2 dt

Guida verificata passo dopo passo
1
Start by rewriting the integral to separate the terms inside the integral: \(\int \frac{1 + \cos 4t}{2} \, dt = \int \frac{1}{2} \, dt + \int \frac{\cos 4t}{2} \, dt\).
Integrate the first term \(\int \frac{1}{2} \, dt\) which is a constant multiple of \(dt\). Recall that \(\int a \, dt = at\) for constant \(a\).
For the second term \(\int \frac{\cos 4t}{2} \, dt\), factor out the constant \(\frac{1}{2}\) to get \(\frac{1}{2} \int \cos 4t \, dt\).
Use the substitution rule or recall the integral formula \(\int \cos (kt) \, dt = \frac{1}{k} \sin (kt) + C\). Apply this with \(k=4\) to find the antiderivative of \(\cos 4t\).
Combine the results from both integrals and add the constant of integration \(C\) to write the most general antiderivative.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
2m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Indefinite Integral and Antiderivative

An indefinite integral represents the most general antiderivative of a function, including a constant of integration. It reverses differentiation, providing a family of functions whose derivative equals the integrand.
Video consigliato:
Percorso guidato
05:04
Introduction to Indefinite Integrals

Integration of Trigonometric Functions

Integrating trigonometric functions like cosine involves using known integral formulas, such as ∫cos(ax) dx = (1/a) sin(ax) + C. Recognizing and applying these formulas simplifies finding antiderivatives.
Video consigliato:
Percorso guidato
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 integral and helps adjust any initial guesses.
Video consigliato:
Percorso guidato
05:53
Finding Differentials
Pratica correlata
Domanda del libro di testo

The intensity of illumination at any point from a light source is proportional to the square of the reciprocal of the distance between the point and the light source. Two lights, one having an intensity eight times that of the other, are 6 m apart. How far from the stronger light is the total illumination least?

250
views
Domanda del libro di testo

Identifying Extrema


In Exercises 19–40:


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


b. Identify the function’s local extreme values, if any, saying where they occur.


f(x) = x¹ᐟ³(x + 8)

183
views
Domanda del libro di testo

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

__________

y = √ 3 + 2𝓍 ―𝓍²

226
views
Domanda del libro di testo

Initial Value Problems


Solve the initial value problems in Exercises 71–90.


d²y/dx² = 2 − 6x; y′(0) = 4, y(0) = 1

23
views
Domanda del libro di testo

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?

188
views
Domanda del libro di testo

Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.

80. y' = 1 - cot²θ, for 0 < θ < π

165
views