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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.6.16
Chapter 8, Problem 8.6.16

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ e^(-3t) sin(4t) dt

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1
Recognize that the integral is of the form \(\int e^{at} \sin(bt) \, dt\), where \(a = -3\) and \(b = 4\). This is a standard integral that can be found in the table of integrals.
Recall the formula for the integral: \(\int e^{at} \sin(bt) \, dt = \frac{e^{at}}{a^2 + b^2} (a \sin(bt) - b \cos(bt)) + C\), where \(C\) is the constant of integration.
Substitute the values \(a = -3\) and \(b = 4\) into the formula to express the integral in terms of \(t\).
Write the integral as \(\int e^{-3t} \sin(4t) \, dt = \frac{e^{-3t}}{(-3)^2 + 4^2} (-3 \sin(4t) - 4 \cos(4t)) + C\).
Simplify the denominator and the expression inside the parentheses as much as possible to get the final integral expression.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Integration of Products of Exponential and Trigonometric Functions

Integrals involving products of exponential and trigonometric functions often require special techniques such as integration by parts or using known integral formulas. Recognizing the form helps in applying the correct formula or method to simplify the integral efficiently.
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Use of Integral Tables

Integral tables provide pre-calculated formulas for common integrals, saving time and effort. Knowing how to locate and apply the correct formula from the table is essential, especially for integrals involving combinations of exponential and trigonometric functions.
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Integration Using Partial Fractions

Integration by Parts

Integration by parts is a technique based on the product rule for differentiation, useful for integrating products of functions. It involves choosing parts of the integrand as 'u' and 'dv' to simplify the integral, often applied repeatedly for integrals like ∫ e^(at) sin(bt) dt.
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Integration by Parts for Definite Integrals
Related Practice
Textbook Question

Volume of water in a swimming pool

A rectangular swimming pool is 30 ft wide and 50 ft long. The accompanying table shows the depth h(x) of the water at 5-ft intervals from one end of the pool to the other. Estimate the volume of water in the pool using the Trapezoidal Rule with n = 10 applied to the integral

V = ∫ from 0 to 50 of 30 · h(x) dx.

Textbook Question

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.

∫ x² tan⁻¹(x / 2) dx

Textbook Question

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫ ((2ˣ - 1) / 3ˣ) dx

Textbook Question

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

∫₋∞^∞ (x dx) / (x² + 4)^(3/2)

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Textbook Question

Exercises 59–64 require the use of various trigonometric identities before you evaluate the integrals.

∫ sin(θ) sin(2θ) sin(3θ) dθ

Textbook Question

Evaluate the integrals in Exercises 1–24 using integration by parts.

∫ p⁴ e^(-p) dp