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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 8.1.12
Capitolo 8, Problema 8.1.12

7–64. Integration review Evaluate the following integrals.
12. ∫ from -5 to 0 of dx / √(4 - x)

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Step 1: Recognize the integral ∫ dx / √(4 - x) as a definite integral with limits of integration from -5 to 0. The integrand involves a square root, which suggests a substitution method might be helpful.
Step 2: Let u = 4 - x. Then, differentiate u with respect to x to find du/dx = -1, or equivalently, dx = -du. Substitute u into the integral to replace x and dx.
Step 3: Adjust the limits of integration according to the substitution. When x = -5, u = 4 - (-5) = 9. When x = 0, u = 4 - 0 = 4. The integral now becomes ∫ from u=9 to u=4 of -du / √u.
Step 4: Simplify the integral using the negative sign and rewrite it as -∫ from u=9 to u=4 of du / √u. Recognize that 1/√u can be expressed as u^(-1/2), which is a standard power rule for integration.
Step 5: Apply the power rule for integration to u^(-1/2). The integral of u^(-1/2) is 2√u. Evaluate this antiderivative at the new limits of integration (u=9 and u=4), and subtract the results to find the value of the definite integral.

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Definite Integrals

A definite integral represents the signed area under a curve between two specified limits. It is denoted as ∫ from a to b of f(x) dx, where 'a' and 'b' are the lower and upper limits, respectively. The result of a definite integral is a numerical value that quantifies the accumulation of the function's values over the interval [a, b].
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Definition of the Definite Integral

Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and recognizing standard forms. For the integral ∫ dx / √(4 - x), a substitution method can simplify the expression, making it easier to evaluate the integral.
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Integration by Parts for Definite Integrals

Square Root Functions

Square root functions, such as √(4 - x), are important in calculus as they often appear in integrals and derivatives. Understanding their behavior, including their domain and range, is crucial for evaluating integrals involving them. In this case, the expression under the square root must be non-negative, which influences the limits of integration.
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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.

15. ∫ x / √(4x + 1) dx

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95–98. {Use of Tech} Numerical methods Use numerical methods or a calculator to approximate the following integrals as closely as possible. The exact value of each integral is given.

98. ∫(from 0 to 1) (ln x)/(1+x) dx = -π²/12

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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.

31. ∫ √(x² - 8x) dx, x > 8

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Domanda del libro di testo

63. (Use of Tech) Normal distribution of heights

The heights of U.S. men are normally distributed with a mean of 69 in and a standard deviation of 3 in. This means that the fraction of men with a height between a and b (with a < b) inches is given by the integral

(1/(3√(2π))) ∫ₐᵇ e^(-((x-69)/3)²/2) dx.

What percentage of American men are between 66 and 72 inches tall? Use the method of your choice, and experiment with the number of subintervals until you obtain successive approximations that differ by less than 10⁻³.

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9–61. Trigonometric integrals Evaluate the following integrals.

16. ∫ sin²θ cos⁵θ dθ

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76–83. Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals.

82. ∫ [dx / (x√(1 + 2x))]

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