Textbook Question
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
85. lim(x→1) (x² + 3x - 4)/(x - 1)
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.PE.41
Evaluate the integrals in Exercises 31–78.
41. ∫(from 0 to 4)2t/(t² - 25)dt
Verified step by step guidance1
Identify the integral to be evaluated: \(\int_0^4 \frac{2t}{t^2 - 25} \, dt\).
Notice that the numerator \$2t\( is the derivative of the denominator \)t^2 - 25$, which suggests using the substitution method.
Let \(u = t^2 - 25\), then compute \(du = 2t \, dt\). This means \(2t \, dt = du\).
Change the limits of integration according to the substitution: when \(t=0\), \(u = 0^2 - 25 = -25\); when \(t=4\), \(u = 4^2 - 25 = 16 - 25 = -9\).
Rewrite the integral in terms of \(u\): \(\int_{-25}^{-9} \frac{1}{u} \, du\), which can be integrated using the natural logarithm function.
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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 specified limits. It is represented as ∫ from a to b of f(t) dt, where a and b are the lower and upper bounds. Evaluating definite integrals involves finding the antiderivative and then applying the Fundamental Theorem of Calculus.
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Integration Techniques - Substitution
Substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable, which transforms the integral into a simpler form. This technique is especially useful when the integrand contains a function and its derivative.
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Handling Rational Functions in Integration
Rational functions are ratios of polynomials, and integrating them often requires algebraic manipulation like partial fraction decomposition or substitution. Recognizing the structure of the numerator and denominator helps in choosing the right method to simplify and evaluate the integral.
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Related Practice
Textbook Question
133. What is the age of a sample of charcoal in which 90% of the carbon-14 originally present has decayed?
Textbook Question
In Exercises 129–132 solve the initial value problem.
129. dy/dx = e^(-x-y-2), y(0) = -2
Textbook Question
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
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Textbook Question
113. The function f(x) = e^x + x, being differentiable and one-to-one, has a differentiable inverse f^(-1)(x). Find the value of df^(-1)/dx at the point f (ln 2).
Textbook Question
109. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
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