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.PE.73
Capitolo 4, Problema 4.PE.73

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.

∫ (𝓍³ + 5𝓍 ―7) d𝓍

Guida verificata passo dopo passo
1
Identify the integral to solve: \(\int (x^{3} + 5x - 7) \, dx\).
Recall the power rule for integration: \(\int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C\), where \(n \neq -1\).
Apply the power rule to each term separately: integrate \(x^{3}\), \$5x$, and $-7$ individually.
For \(x^{3}\), the integral is \(\frac{x^{4}}{4}\); for \$5x$, treat the constant 5 as a multiplier and integrate $x$ to get \(\frac{x^{2}}{2}\), so the term becomes \(5 \times \frac{x^{2}}{2}\); for $-7$, integrate the constant to get $-7x$.
Combine all integrated terms 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:
1m

Concetti chiave

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

Indefinite Integral

An indefinite integral represents the most general antiderivative of a function, expressed as a family of functions plus a constant of integration (C). It reverses differentiation and is written without limits, indicating all possible antiderivatives.
Video consigliato:
Percorso guidato
05:04
Introduction to Indefinite Integrals

Power Rule for Integration

The power rule states that the integral of x^n (where n ≠ -1) is (x^(n+1)) / (n+1) + C. This rule is essential for integrating polynomial terms like x³ and x, allowing straightforward calculation of their antiderivatives.
Video consigliato:
Percorso guidato
04:04
Power Rule for Indefinite Integrals

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.
Video consigliato:
Percorso guidato
05:53
Finding Differentials
Pratica correlata
Domanda del libro di testo

Initial Value Problems

Solve the initial value problems in Exercises 89–92.


dy/dx = (𝓍 + 1/𝓍)² , y(1)= 1

23
views
Domanda del libro di testo

Initial Value Problems

Solve the initial value problems in Exercises 89–92.

d^3 r/dt^3 = - cos t; r''(0) = r'(0) = 0 , r(0) = -1

16
views
Domanda del libro di testo

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

34
views
Domanda del libro di testo

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

20
views
Domanda del libro di testo

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.

∫ 𝓍³ (1 + 𝓍⁴ )⁻¹/⁴ d𝓍

17
views
Domanda del libro di testo

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

22
views