Skip to main content
Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 5 - IntegrationProblema 5.5.8
Capitolo 5, Problema 5.5.8

Use the given substitution to evaluate the following indefinite integrals. Check your answer by differentiating.                                                                                              
                                                                                                                                                                                       
 ∫ 8𝓍 cos (4𝓍² + 3) d𝓍, u = 4𝓍² + 3

Guida verificata passo dopo passo
1
Step 1: Identify the substitution variable. The problem suggests using u = 4𝓍² + 3. This substitution simplifies the integral by replacing the complex expression inside the cosine function.
Step 2: Compute the derivative of u with respect to 𝓍 to find du. Differentiating u = 4𝓍² + 3 gives du/d𝓍 = 8𝓍. Therefore, du = 8𝓍 d𝓍.
Step 3: Rewrite the integral in terms of u. Substitute u = 4𝓍² + 3 and du = 8𝓍 d𝓍 into the integral. The integral becomes ∫ cos(u) du.
Step 4: Solve the simplified integral. The integral of cos(u) with respect to u is sin(u) + C, where C is the constant of integration.
Step 5: Substitute back the original variable. Replace u with 4𝓍² + 3 to express the solution in terms of 𝓍. The final result is sin(4𝓍² + 3) + 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:
1m

Concetti chiave

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

Substitution Method

The substitution method is a technique used in integration to simplify the process by replacing a complex expression with a single variable. In this case, we let u = 4x² + 3, which transforms the integral into a more manageable form. This method is particularly useful when dealing with integrals involving composite functions, as it allows for easier integration and ultimately leads to the solution.
Video consigliato:
07:33
Euler's Method

Differentiation

Differentiation is the process of finding the derivative of a function, which measures how the function changes as its input changes. In the context of checking the result of an integral, differentiating the antiderivative obtained from the integration process should yield the original integrand. This step is crucial for verifying the correctness of the integration and ensuring that no mistakes were made during the substitution or integration steps.
Video consigliato:
Percorso guidato
05:53
Finding Differentials

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed with a constant of integration (C) because there are infinitely many antiderivatives differing by a constant. Understanding indefinite integrals is essential for solving problems in calculus, as they provide the foundational concept for finding areas under curves and solving differential equations.
Video consigliato:
Percorso guidato
05:04
Introduction to Indefinite Integrals
Pratica correlata
Domanda del libro di testo

Symmetry of composite functions Prove that the integrand is either even or odd. Then give the value of the integral or show how it can be simplified. Assume f and g are even functions and p and q are odd functions.

∫ᵃ₋ₐ ƒ(p(𝓍)) d𝓍

110
views
Domanda del libro di testo

Symmetry in integrals Use symmetry to evaluate the following integrals.

∫²₋₂ (x² + x³) dx

113
views
Domanda del libro di testo

Left and right Riemann sums Use the figures to calculate the left and right Riemann sums for f on the given interval and for the given value of n.

ƒ(𝓍) = x + 1 on [1,6] ; n = 5

533
views
Domanda del libro di testo

Approximating area from a graph Approximate the area of the region bounded by the graph (see figure) and the 𝓍-axis by dividing the interval [1, 7] into n = 6 subintervals. Use a left and right Riemann sum to obtain two different approximations.                                                                                                                                                                         

74
views
Domanda del libro di testo

Symmetry in integrals Use symmetry to evaluate the following integrals.

∫²⁰⁰₋₂₀₀ 2x⁵ dx

109
views
Domanda del libro di testo

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus


∫¹₁/₂ (t⁻³ ― 8) dt

58
views