Ch. 8 - Integration Techniques

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 8 - Integration Techniques

Problema 8.2.79

Capitolo 8, Problema 8.2.79

79. Tabular integration extended Refer to Exercise 77.
a. The following table shows the method of tabular integration applied to
∫ eˣ cos x dx.

Use the table to express ∫ eˣ cos x dx in terms of the sum of functions and an indefinite integral.
b. Solve the equation in part (a) for ∫ eʳ cos z dz.
c. Evaluate ∫ e⁻ᶻ sin 3z dz by applying the idea from parts (a) and (b).

Guida verificata passo dopo passo

Identify the functions from the table: let \( f(x) = e^x \) and \( g(x) = \cos x \). The table shows derivatives of \( f \) and integrals of \( g \) along with signs for tabular integration.

Use the tabular integration method to write \( \int e^x \cos x \, dx \) as the sum of products of \( f \) and integrals of \( g \) with alternating signs, plus the remaining integral. From the table, this gives: \[ \int e^x \cos x \, dx = + e^x \sin x - e^x (-\cos x) + \int e^x (-\cos x) \, dx \]

Simplify the expression inside the integral and the sum of functions: \[ \int e^x \cos x \, dx = e^x \sin x + e^x \cos x - \int e^x \cos x \, dx \]

Recognize that the integral \( \int e^x \cos x \, dx \) appears on both sides of the equation. To solve for it, add \( \int e^x \cos x \, dx \) to both sides to isolate the integral term.

Divide both sides of the resulting equation by 2 to solve explicitly for \( \int e^x \cos x \, dx \). This gives the integral in terms of elementary functions without any remaining integral.

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Tabular Integration Method

Tabular integration is a streamlined technique for repeated integration by parts, especially useful when differentiating one function repeatedly leads to a simple pattern. It involves creating a table of derivatives of one function and integrals of another, then combining these with alternating signs to find the integral efficiently.

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Tabular Integration by Parts Example 6

Integration by Parts

Integration by parts is a fundamental technique based on the product rule for differentiation. It transforms the integral of a product of functions into simpler integrals, using the formula ∫u dv = uv - ∫v du. This method is essential for integrals involving products of exponential, trigonometric, or polynomial functions.

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Integration by Parts for Definite Integrals

Solving Integral Equations Involving the Same Integral

Sometimes, applying integration by parts leads to an equation containing the original integral on both sides. Solving such equations involves algebraic manipulation to isolate the integral, allowing evaluation of integrals like ∫eˣ cos x dx, which recur during the integration process.

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Integrals Involving Natural Logs: Substitution

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54–57. {Use of Tech} Comparing the Midpoint and Trapezoid Rules Compare the errors in the Midpoint and Trapezoid Rules with n = 4, 8, 16, and 32 subintervals when they are applied to the following integrals (with their exact values given).

59. ∫(from 0 to π) ln(5 + 3cosx) dx = π ln(9/2)

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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.

6. (4x + 1)/(4x² - 1)

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85. Another form of ∫ sec x dx

a. Verify the identity:

sec x = cos x / (1 - sin² x)

b. Use the identity in part (a) to verify that:

∫ sec x dx = (1/2) ln |(1 + sin x)/(1 - sin x)| + C

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23-64. Integration Evaluate the following integrals.

60.∫ 1/[(y² + 1)(y² + 2)] dy