Skip to main content
Ch. 8 - Integration Techniques
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. 8 - Integration TechniquesProblema 8.6.7
Capitolo 8, Problema 8.6.7

7–84. Evaluate the following integrals.
7. ∫ from 0 to π/2 [sin θ / (1 + cos² θ)] dθ

Guida verificata passo dopo passo
1
Step 1: Recognize that the integral involves trigonometric functions. The integrand is sin(θ) / (1 + cos²(θ)). To simplify, consider substitution methods or trigonometric identities.
Step 2: Use the substitution u = cos(θ). Then, compute du = -sin(θ)dθ. This substitution will transform the integral into terms of u, making it easier to evaluate.
Step 3: Change the limits of integration according to the substitution. When θ = 0, u = cos(0) = 1. When θ = π/2, u = cos(π/2) = 0. The integral now has limits from u = 1 to u = 0.
Step 4: Rewrite the integral in terms of u. The original integral becomes ∫ from 1 to 0 [-du / (1 + u²)]. Notice the negative sign from du = -sin(θ)dθ.
Step 5: Reverse the limits of integration to remove the negative sign, resulting in ∫ from 0 to 1 [du / (1 + u²)]. This integral is now in a standard form, which can be evaluated using the arctangent function: ∫ [du / (1 + u²)] = arctan(u).

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.

Definite Integrals

A definite integral calculates the accumulation of a quantity, represented as the area under a curve, between two specified limits. In this case, the integral is evaluated from 0 to π/2, meaning we are interested in the area under the curve of the function sin(θ) / (1 + cos²(θ)) from θ = 0 to θ = π/2.
Video consigliato:
Percorso guidato
05:43
Definition of the Definite Integral

Trigonometric Functions

Trigonometric functions, such as sine and cosine, relate angles to ratios of sides in right triangles. Understanding these functions is crucial for evaluating integrals involving them, as they often appear in calculus problems. The integral in question involves sin(θ) and cos²(θ), which are fundamental in trigonometric identities and integration techniques.
Video consigliato:
Percorso guidato
6:04
Introduction to Trigonometric Functions

Integration Techniques

Integration techniques are methods used to find the integral of a function, which can include substitution, integration by parts, or recognizing standard forms. For the given integral, recognizing the structure of the integrand and possibly using a substitution can simplify the evaluation process, making it easier to compute the area under the curve.
Video consigliato:
Percorso guidato
06:18
Integration by Parts for Definite Integrals
Pratica correlata
Domanda del libro di testo

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

21. ∫ cos x / (sin² x + 2 sin x) dx

52
views
Domanda del libro di testo

7–84. Evaluate the following integrals.

77. ∫ arccosx dx

64
views
Domanda del libro di testo

9–61. Trigonometric integrals Evaluate the following integrals.

26. ∫ sin³x cos³ᐟ²x dx

65
views
Domanda del libro di testo

58–61. {Use of Tech} Using Simpson's Rule Approximate the following integrals using Simpson's Rule. Experiment with values of n to ensure the error is less than 10⁻³.

60. ∫(from 0 to π) ln(2 + cos x) dx = π ln((2 + √3)/2)

83
views
Domanda del libro di testo

9–61. Trigonometric integrals Evaluate the following integrals.

43. ∫ tan³(4x) dx

73
views
Domanda del libro di testo

49–63. {Use of Tech} Integrating with a CAS Use a computer algebra system to evaluate the following integrals. Find both an exact result and an approximate result for each definite integral. Assume a is a positive real number.

61. ∫₀¹ (ln x) ln(1 + x) dx

56
views