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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.RE.4
Chapter 7, Problem 7.RE.4

2–9. Integrals Evaluate the following integrals.


∫₁⁴ (10^{√x} / √x) dx

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Identify the integral to be solved: \(\int_1^4 \frac{10^{\sqrt{x}}}{\sqrt{x}} \, dx\).
Use the substitution method by letting \(t = \sqrt{x}\), which implies \(x = t^2\).
Calculate the differential \(dx\) in terms of \(dt\): since \(x = t^2\), then \(dx = 2t \, dt\).
Rewrite the integral in terms of \(t\): substitute \(\sqrt{x} = t\) and \(dx = 2t \, dt\) into the integral, and adjust the limits accordingly (when \(x=1\), \(t=1\); when \(x=4\), \(t=2\)).
Simplify the integral expression after substitution and set it up for integration with respect to \(t\).

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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 points, called limits of integration. It results in a numerical value representing the accumulation of quantities, such as area or total change, over the interval.
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Definition of the Definite Integral

Substitution Method

The substitution method simplifies integrals by changing variables to transform a complex integral into a more manageable form. It involves identifying a part of the integrand as a new variable and rewriting the integral in terms of this variable.
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Euler's Method

Exponential Functions with Variable Exponents

Exponential functions where the exponent is a function of the variable, like 10^{√x}, require careful handling during integration. Understanding how to differentiate and integrate such functions often involves using substitution and properties of logarithms.
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a. Use an exponential model to estimate the population in 20 years. Assume the annual growth rate is constant.

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Textbook Question

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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

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.

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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