Textbook Question
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
9. ∫[5 to 5√3] √(100 - x²) dx
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 8.2.32
9–40. Integration by parts Evaluate the following integrals using integration by parts.
32. ∫ from 0 to 1 x² 2ˣ dx
Verified step by step guidance1
Identify the integral to solve: \(\int_0^1 x^2 2^x \, dx\).
Choose functions for integration by parts. Let \(u = x^2\) (which simplifies when differentiated) and \(dv = 2^x \, dx\) (which can be integrated).
Compute \(du\) and \(v\): differentiate \(u\) to get \(du = 2x \, dx\), and integrate \(dv\) to find \(v\). Recall that \(\int a^x \, dx = \frac{a^x}{\ln a} + C\), so here \(v = \frac{2^x}{\ln 2}\).
Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\). Substitute the expressions for \(u\), \(v\), \(du\) to write the integral in terms of a simpler integral.
Evaluate the resulting integral and apply the limits from 0 to 1. If the new integral still requires integration by parts, repeat the process until the integral is fully evaluated.
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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 u and dv wisely simplifies the integral evaluation.
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Integration by Parts for Definite Integrals
Exponential Functions
Exponential functions like 2^x grow or decay at rates proportional to their value. When integrating, it’s important to recognize that the derivative and integral of 2^x involve the natural logarithm of the base, since d/dx(2^x) = 2^x ln(2).
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6:13Exponential Functions
Definite Integrals
Definite integrals compute the net area under a curve between two limits. After finding the antiderivative, evaluate it at the upper and lower bounds and subtract to find the integral’s value over the interval [0,1].
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05:43Definition of the Definite Integral
Related Practice
Textbook Question
37-40. {Use of Tech} Temperature data
Howdy temperature data for Boulder, Colorado; San Francisco, California; Nantucket, Massachusetts; and Duluth, Minnesota, over a 12-hr period on the same day of January are shown in the figure.
Assume these data are taken from a continuous temperature function T(t). The average temperature (in °F) over the 12-hr period is:
T_avg = (1/12) × ∫(0 to 12) T(t) dt
38. Find an accurate approximation to the average temperature over the 12-hr period for San Francisco. State your method.
Textbook Question
9–40. Integration by parts Evaluate the following integrals using integration by parts.
17. ∫ x · 3x dx
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Textbook Question
102–106. Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by F(s) = ∫[0 to ∞] e^(-st) f(t) dt, where we assume s is a positive real number. For example, to find the Laplace transform of f(t) = e^(-t), the following improper integral is evaluated using integration by parts:
F(s) = ∫[0 to ∞] e^(-st) e^(-t) dt = ∫[0 to ∞] e^(-(s+1)t) dt = 1/(s+1).
Verify the following Laplace transforms, where a is a real number.
104. f(t) = t → F(s) = 1/s²
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Textbook Question
69. Different substitutions
b. Evaluate ∫(tan x sec² x) dx using the substitution u=secx.
Textbook Question
49–52. {Use of Tech} Simpson’s Rule
Apply Simpson’s Rule to the following integrals. It is easiest to obtain the Simpson’s Rule approximations from the Trapezoid Rule approximations, as in Example 8. Make a table similar to Table 8.8 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error.
51. ∫(from 0 to π) e⁻ᵗ sin(t) dt = ½(e⁻ᵖⁱ + 1)