Domanda del libro di testo
Evaluate the following integrals.
65. ∫ from 0 to 1/6 1/√(1 - 9x²) dx
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 8, Problema 8.6.77
7–84. Evaluate the following integrals.
77. ∫ arccosx dx
Guida verificata passo dopo passo1
Identify the integral to solve: \(\int \arccos x \, dx\).
Use integration by parts, which states: \(\int u \, dv = uv - \int v \, du\). Choose \(u = \arccos x\) and $dv = dx$.
Compute \(du\) by differentiating \(u\): since \(u = \arccos x\), then \(du = -\frac{1}{\sqrt{1 - x^2}} \, dx\). Also, integrate \(dv\) to get \(v = x\).
Apply the integration by parts formula: \(\int \arccos x \, dx = x \arccos x - \int x \left(-\frac{1}{\sqrt{1 - x^2}}\right) dx = x \arccos x + \int \frac{x}{\sqrt{1 - x^2}} \, dx\).
Evaluate the remaining integral \(\int \frac{x}{\sqrt{1 - x^2}} \, dx\) by using a substitution such as \(w = 1 - x^2\), then express the integral in terms of \(w\) and solve.
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Integration by Parts
Integration by parts is a technique used to integrate products of functions. It is based on the product rule for differentiation and follows the formula ∫u dv = uv - ∫v du. Choosing appropriate u and dv simplifies the integral, especially when dealing with inverse trigonometric functions.
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Percorso guidato
06:18
Integration by Parts for Definite Integrals
Derivative of the Inverse Cosine Function
The derivative of arccos(x) is -1 / √(1 - x²). Knowing this derivative is essential when applying integration by parts, as it helps determine du when u = arccos(x). This derivative reflects the rate of change of the inverse cosine function.
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07:26Derivatives of Inverse Sine & Inverse Cosine
Basic Integration Techniques
Understanding fundamental integration methods, such as integrating powers and roots, is important for solving the resulting integrals after applying integration by parts. This includes recognizing standard integral forms and manipulating expressions to fit these forms.
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Percorso guidato
06:07Basic Rules 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
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Domanda del libro di testo
7–84. Evaluate the following integrals.
7. ∫ from 0 to π/2 [sin θ / (1 + cos² θ)] dθ
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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)
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Domanda del libro di testo
9–61. Trigonometric integrals Evaluate the following integrals.
43. ∫ tan³(4x) dx
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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
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