Textbook Question
2–9. Integrals Evaluate the following integrals.
∫₁⁴ (10^{√x} / √x) dx
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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
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
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Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.RE.9
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.RE.9Chapter 7, Problem 7.RE.9
2–9. Integrals Evaluate the following integrals.
∫₀¹ (x² / (9 − x⁶)) dx
Verified step by step guidance1
Identify the integral to be evaluated: \(\int_0^1 \frac{x^2}{9 - x^6} \, dx\).
Look for a substitution that simplifies the denominator. Notice that the denominator is \(9 - x^6\), and the numerator is \(x^2\). Consider substituting \(u = x^3\) because \(x^6 = (x^3)^2 = u^2\).
Compute the differential \(du\): since \(u = x^3\), then \(du = 3x^2 \, dx\), which implies \(x^2 \, dx = \frac{du}{3}\).
Rewrite the integral in terms of \(u\): change the limits accordingly. When \(x=0\), \(u=0^3=0\); when \(x=1\), \(u=1^3=1\). The integral becomes \(\int_0^1 \frac{1}{9 - u^2} \cdot \frac{du}{3}\).
Simplify the integral to \(\frac{1}{3} \int_0^1 \frac{1}{9 - u^2} \, du\). This is a standard integral that can be solved using partial fractions or recognizing it as a form related to inverse hyperbolic functions or logarithms.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integrals
A definite integral calculates the net area under a curve between two specific limits. It is represented as ∫_a^b f(x) dx, where a and b are the lower and upper bounds. Evaluating definite integrals often involves finding an antiderivative and then applying the Fundamental Theorem of Calculus.
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Definition of the Definite Integral
Substitution Method
The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. It involves choosing a substitution u = g(x) such that the integral in terms of u is easier to evaluate. This technique is especially useful when the integrand contains composite functions.
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Handling Rational Functions with Polynomial Denominators
Integrals involving rational functions with polynomial denominators often require algebraic manipulation or substitution to simplify. Recognizing patterns, such as powers in numerator and denominator, helps in choosing an appropriate substitution or partial fraction decomposition to evaluate the integral.
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Related Practice
Textbook Question
Moore’s Law In 1965, Gordon Moore observed that the number of transistors that could be placed on an integrated circuit was approximately doubling each year, and he predicted that this trend would continue for another decade. In 1975, Moore revised the doubling time to every two years, and this prediction became known as Moore’s Law.
a. In 1979, Intel introduced the Intel 8088 processor; each of its integrated circuits contained 29,000 transistors. Use Moore’s revised doubling time to find a function y(t) that approximates the number of transistors on an integrated circuit t years after 1979.
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Textbook Question
2–9. Integrals Evaluate the following integrals.
∫ dx / √(x² − 9),x > 3
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Textbook Question
Caffeine An adult consumes an espresso containing 75 mg of caffeine. If the caffeine has a half-life of 5.5 hours, when will the amount of caffeine in her bloodstream equal 30 mg?
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Textbook Question
Population growth The population of a large city grows exponentially with a current population of 1.3 million and a predicted population of 1.45 million 10 years from now.
b. Find the doubling time of the population.
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Textbook Question
Savings account A savings account advertises an annual percentage yield (APY) of 5.4%, which means that the balance in the account increases at an annual growth rate of 5.4%/yr.
b. What is the doubling time of the balance?
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