Textbook Question
7–58. Improper integrals Evaluate the following integrals or state that they diverge.
22. ∫ (from -∞ to -2) (1/x²) sin(π/2) dx
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- 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: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.2.29
9–40. Integration by parts Evaluate the following integrals using integration by parts.
29. ∫ e⁻ˣ sin(4x) dx
Verified step by step guidance1
Identify the integral to solve: \(\int e^{-x} \sin(4x) \, dx\).
Recall the integration by parts formula: \(\int u \, dv = uv - \int v \, du\).
Choose \(u\) and \(dv\) wisely. For this integral, let \(u = \sin(4x)\) and \(dv = e^{-x} dx\).
Compute \(du\) and \(v\): differentiate \(u\) to get \(du = 4 \cos(4x) dx\), and integrate \(dv\) to get \(v = -e^{-x}\).
Apply the integration by parts formula: substitute \(u\), \(v\), \(du\) into \(\int u \, dv = uv - \int v \, du\), then simplify the resulting integral. You may need to apply integration by parts a second time or solve for the original integral algebraically.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration by Parts
Integration by parts is a technique derived from the product rule of differentiation. It transforms the integral of a product of functions into simpler integrals using the formula ∫u dv = uv - ∫v du. Choosing appropriate u and dv is crucial to simplify the integral effectively.
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Integration by Parts for Definite Integrals
Integration of Exponential and Trigonometric Functions
Integrals involving products of exponential and trigonometric functions often require repeated application of integration by parts. Recognizing patterns and using algebraic manipulation helps to solve these integrals, especially when the integral reappears during the process.
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05:11Integrals of General Exponential Functions
Solving for the Original Integral
When integration by parts leads to an integral expression containing the original integral, algebraic techniques are used to isolate and solve for it. This method is common in integrals involving products of exponential and trigonometric functions, enabling the evaluation of otherwise complex integrals.
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5:59Graph Hyperbolas NOT at the Origin
Related Practice
Textbook Question
72. Between the sine and inverse sine Find the area of the region bound by the curves y = sin x and y = sin⁻¹x on the interval [0, 1/2].
Textbook Question
60–69. Completing the square Evaluate the following integrals.
65. ∫[1/2 to (√2 + 3)/(2√2)] dx / (8x² - 8x + 11)
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Textbook Question
90. Work Let R be the region in the first quadrant bounded by the curve y = √(x⁴ - 4)
and the lines y = 0 and y = 2. Suppose a tank that is full of water has the shape of a solid of revolution obtained by revolving region R about the y-axis. How much work is required to pump all the water to the top of the tank? Assume x and y are in meters.
Textbook Question
67-70. Integrals of the form ∫ sin(mx)cos(nx) dx Use the following product-to-sum identities to evaluate the given integrals:
sin(mx)sin(nx) = ½[cos((m-n)x) - cos((m+n)x)]
sin(mx)cos(nx) = ½[sin((m-n)x) + sin((m+n)x)]
cos(mx)cos(nx) = ½[cos((m-n)x) + cos((m+n)x)]
68. ∫ sin(5x)sin(7x) dx
Textbook Question
76. Apparent discrepancy
Three different computer algebra systems give the following results:
∫ (dx / (x√(x⁴ − 1))) = ½ cos⁻¹(√(x⁻⁴)) = ½ cos⁻¹(x⁻²) = ½ tan⁻¹(√(x⁴ − 1)).
Explain how all three can be correct.
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