Textbook Question
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
68. Let f(x) = e^(x²).
a. Find a Trapezoid Rule approximation to ∫[0 to 1] e^(x²) dx using n = 50 subintervals.
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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 8.2.60a
60. Two Methods
a. Evaluate ∫(x · ln(x²)) dx using the substitution u = x² and evaluating ∫(ln(u)) du.
Verified step by step guidance1
Start with the integral \( \int x \cdot \ln(x^2) \, dx \). The goal is to use the substitution \( u = x^2 \).
Compute the differential \( du \) from \( u = x^2 \). Since \( \frac{du}{dx} = 2x \), we have \( du = 2x \, dx \), which implies \( x \, dx = \frac{du}{2} \).
Rewrite the integral in terms of \( u \): \( \int x \cdot \ln(x^2) \, dx = \int \ln(u) \cdot x \, dx = \int \ln(u) \cdot \frac{du}{2} = \frac{1}{2} \int \ln(u) \, du \).
Focus on evaluating \( \int \ln(u) \, du \). Use integration by parts where you let \( v = \ln(u) \) and \( dw = du \). Then, \( dv = \frac{1}{u} du \) and \( w = u \).
Apply integration by parts formula: \( \int v \, dw = vw - \int w \, dv \). Substitute the expressions to get \( u \ln(u) - \int u \cdot \frac{1}{u} du = u \ln(u) - \int 1 \, du = u \ln(u) - u + C \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration by Substitution
Integration by substitution is a technique used to simplify integrals by changing variables. It involves substituting a part of the integral with a new variable, making the integral easier to evaluate. In this problem, substituting u = x² transforms the integral into one involving ln(u), which is simpler to handle.
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04:27
Substitution With an Extra Variable
Properties of Logarithms
Understanding logarithmic properties is essential when dealing with integrals involving ln(x²). The property ln(x²) = 2 ln(x) can simplify expressions before integration. Recognizing how to manipulate logarithmic functions helps in rewriting the integral in a more manageable form.
Recommended video:
05:36Change of Base Property
Integration of Logarithmic Functions
Integrating functions involving logarithms, such as ∫ln(u) du, requires specific techniques like integration by parts. Knowing the formula ∫ln(u) du = u ln(u) - u + C is useful. This concept is crucial for completing the integral after substitution.
Recommended video:
5:26Graphs of Logarithmic Functions
Related Practice
Textbook Question
91. [Use of Tech] Regions bounded by exponentials Let a > 0 and let R be the region bounded by the graph of y = e^(-a·x) and the x-axis
on the interval [b, ∞).
a. Find A(a,b), the area of R as a function of a and b.
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Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. ∫(3/(x² + 4)) dx = ∫(3/x²) dx + ∫(3/4) dx.
Textbook Question
Arc length of a parabola Let L(c) be the length of the parabola f(x) = x² from x = 0 to x = c, where c ≥ 0 is a constant.
a. Find an expression for L.
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Textbook Question
57. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. If x = 4 tanθ, then cscθ = 4/x.
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. Suppose ∫_a^b f(x) dx is approximated with Simpson’s Rule using n = 18 subintervals, where |f^(4)(x)| ≤ 1 on [a, b]. The absolute error E_S in approximating the integral satisfies E_S ≤ (Δx)^5 / 10.