Domanda del libro di testo
5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.
15. x / ((x⁴ - 16)²)
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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 8.4.74
{Use of Tech} Using the integral of sec³u By reduction formula 4 in Section 8.3,
∫sec³u du = 1/2 (sec u tan u + ln |sec u + tan u|) + C
Graph the following functions and find the area under the curve on the given interval.
f(x) = 1/(x√(x² - 36)), [12/√3 , 12]
Guida verificata passo dopo passo1
First, rewrite the integral for the area under the curve as \(\int_{12/\sqrt{3}}^{12} \frac{1}{x \sqrt{x^2 - 36}} \, dx\).
Use the substitution \(x = 6 \sec u\) because \(\sqrt{x^2 - 36}\) suggests a secant substitution (since \(36 = 6^2\)). Then, compute \(dx\) in terms of \(du\).
Express the integral entirely in terms of \(u\) by substituting \(x = 6 \sec u\), \(dx = 6 \sec u \tan u \, du\), and \(\sqrt{x^2 - 36} = 6 \tan u\).
Simplify the integral after substitution to get an integral involving \(\sec^3 u\), which matches the form given in the reduction formula: \(\int \sec^3 u \, du\).
Apply the reduction formula \(\int \sec^3 u \, du = \frac{1}{2} (\sec u \tan u + \ln |\sec u + \tan u|) + C\) to evaluate the integral, then substitute back to \(x\) using \(u = \sec^{-1}(x/6)\) and evaluate the definite integral at the given limits.
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Integration Using Reduction Formulas
Reduction formulas simplify complex integrals by expressing them in terms of simpler integrals of the same type. For example, the integral of sec³u can be evaluated using a known reduction formula, which breaks it down into a combination of sec u tan u and a logarithmic term. This technique is essential for handling powers of trigonometric functions.
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08:01
Integration Using Partial Fractions
Definite Integrals and Area Under a Curve
A definite integral calculates the net area between a function's graph and the x-axis over a specified interval. Evaluating ∫ f(x) dx from a to b gives the total area under f(x) between x = a and x = b, accounting for regions above and below the axis. This concept connects integration to geometric interpretation.
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05:43Definition of the Definite Integral
Trigonometric Substitution in Integration
Trigonometric substitution is a method used to simplify integrals involving expressions like √(x² - a²) by substituting x with a trigonometric function (e.g., x = a sec θ). This transforms the integral into a trigonometric integral, which can be easier to evaluate using known identities and formulas, such as the integral of sec³u.
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6:04Introduction to Trigonometric Functions
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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.
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Domanda del libro di testo
Use a substitution to reduce the following integrals to ∫ ln u du. Then evaluate using the formula for ∫ ln x dx.
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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.
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Domanda del libro di testo
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