Ch. 4 - Applications of Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problema 4.7.43

Capitolo 4, Problema 4.7.43

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.

∫(4secx tanx − 2 sec²x)dx

Guida verificata passo dopo passo

Recognize that the integral is of the form \(\int (4 \sec x \tan x - 2 \sec^{2} x) \, dx\), which can be split into two separate integrals: \(\int 4 \sec x \tan x \, dx - \int 2 \sec^{2} x \, dx\).

Recall the standard derivatives: \(\frac{d}{dx}(\sec x) = \sec x \tan x\) and \(\frac{d}{dx}(\tan x) = \sec^{2} x\). This helps identify antiderivatives for each term.

For the first integral, \(\int 4 \sec x \tan x \, dx\), use the fact that the derivative of \(\sec x\) is \(\sec x \tan x\), so the antiderivative is \(4 \sec x\) plus a constant.

For the second integral, \(\int 2 \sec^{2} x \, dx\), use the fact that the derivative of \(\tan x\) is \(\sec^{2} x\), so the antiderivative is \(2 \tan x\) plus a constant.

Combine the results from both integrals and add a single constant of integration \(C\) to write the most general antiderivative: \(4 \sec x - 2 \tan x + C\).

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:

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 form of an antiderivative of a function, including a constant of integration. It reverses differentiation, finding a function whose derivative matches the integrand. Understanding this helps in solving integrals without specified limits.

Video consigliato:

Percorso guidato

05:04

Introduction to Indefinite Integrals

Integration of Trigonometric Functions

Integrating trigonometric functions like secant and tangent requires knowledge of their derivatives and standard integral formulas. Recognizing patterns such as the derivative of sec x being sec x tan x aids in simplifying and solving the integral efficiently.

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 solution and helps identify any errors in the integration process, ensuring the integral is accurate.

Video consigliato:

Percorso guidato

05:53

Finding Differentials

Pratica correlata

Domanda del libro di testo

Roots (Zeros)

Show that the functions in Exercises 19–26 have exactly one zero in the given interval.

r(θ) = 2θ − cos²θ + √2, (−∞, ∞)

178

views

Domanda del libro di testo

Finding Extrema from Graphs

In Exercises 15–20, sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.

g(x) = {−x, 0 ≤ x < 1

x − 1, 1 ≤ x ≤ 2

245

views

Domanda del libro di testo

Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.

5. y=x+sin(2x), -2π/3≤x≤2π/3

317

views

Domanda del libro di testo

Roots (Zeros)

Show that the functions in Exercises 19–26 have exactly one zero in the given interval.

g(t) = 1/(1 − t) + √(1 + t) − 3.1, (−1, 1)

220

views

Domanda del libro di testo

Initial Value Problems

Solve the initial value problems in Exercises 71–90.