Domanda del libro di testo
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
-(1/3)x⁻⁴ᐟ³
39
views
Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 4, Problema 4.7.11c
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
sin πx − 3sin 3x
Guida verificata passo dopo passo1
Recall that an antiderivative (or indefinite integral) of a function \( f(x) \) is a function \( F(x) \) such that \( F'(x) = f(x) \). Our goal is to find \( F(x) \) given \( f(x) = \sin(\pi x) - 3 \sin(3x) \).
Use the basic antiderivative formula for sine: \( \int \sin(ax) \, dx = -\frac{1}{a} \cos(ax) + C \), where \( a \) is a constant and \( C \) is the constant of integration.
Apply this formula to each term separately: For \( \sin(\pi x) \), the antiderivative is \( -\frac{1}{\pi} \cos(\pi x) \). For \( -3 \sin(3x) \), factor out the constant \( -3 \) and integrate \( \sin(3x) \) to get \( -3 \times \left(-\frac{1}{3} \cos(3x)\right) \).
Simplify the expression after integration: The \( -3 \times -\frac{1}{3} \) simplifies to \( +1 \), so the antiderivative of \( -3 \sin(3x) \) is \( + \cos(3x) \).
Combine the results and add the constant of integration \( C \) to write the full antiderivative: \( F(x) = -\frac{1}{\pi} \cos(\pi x) + \cos(3x) + C \).
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:
7mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Antiderivatives (Indefinite Integrals)
An antiderivative of a function is another function whose derivative equals the original function. Finding antiderivatives involves reversing differentiation, often represented as the indefinite integral with a constant of integration. For example, the antiderivative of sin(x) is -cos(x) + C.
Video consigliato:
Percorso guidato
05:04
Introduction to Indefinite Integrals
Integration of Trigonometric Functions
Integrating trigonometric functions like sin(kx) requires using known integral formulas and applying the chain rule in reverse. Specifically, ∫sin(kx) dx = -cos(kx)/k + C, where k is a constant. Recognizing these patterns helps in quickly finding antiderivatives mentally.
Video consigliato:
Percorso guidato
6:04Introduction to Trigonometric Functions
Verification by Differentiation
After finding an antiderivative, differentiating it should return the original function. This step confirms the correctness of the solution. For example, differentiating -cos(πx)/π yields sin(πx), verifying the antiderivative.
Video consigliato:
Percorso guidato
05:53Finding Differentials
Pratica correlata
Domanda del libro di testo
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
√x + 1/√x
29
views
Domanda del libro di testo
Analyzing Functions from Derivatives
Answer the following questions about the functions whose derivatives are given in Exercises 1–14:
c. At what points, if any, does f assume local maximum or minimum values?
f′(x) = x(x − 1)
137
views
Domanda del libro di testo
Analyzing Functions from Derivatives
Answer the following questions about the functions whose derivatives are given in Exercises 1–14:
c. At what points, if any, does f assume local maximum or minimum values?
f′(x) = (x − 1)(x + 2)(x − 3)
256
views
Domanda del libro di testo
Dependence on Initial Point
8. Using the function shown in the figure, and, for each initial estimate x_0, determine graphically what happens to the sequence of Newton’s method approximations
c. x_0=2
194
views
Domanda del libro di testo
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
2 - 5 / x²
33
views