Textbook Question
84. Find lim(x→∞) (√(x² + 1) - √x).
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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 7.2.49
Evaluate the integrals in Exercises 39–56.
49. ∫3sec²t/(6 + 3tan(t)) dt
Verified step by step guidance1
Identify the integral to solve: \(\int \frac{3 \sec^{2} t}{6 + 3 \tan t} \, dt\).
Notice that the denominator contains \(6 + 3 \tan t\), and the numerator contains \(3 \sec^{2} t\). Recall that the derivative of \(\tan t\) is \(\sec^{2} t\), which suggests a substitution.
Let \(u = 6 + 3 \tan t\). Then, compute \(du\): since \(\frac{d}{dt}(\tan t) = \sec^{2} t\), we have \(du = 3 \sec^{2} t \, dt\).
Rewrite the integral in terms of \(u\) and \(du\): the numerator \(3 \sec^{2} t \, dt\) becomes \(du\), and the denominator is \(u\), so the integral becomes \(\int \frac{1}{u} \, du\).
Integrate \(\int \frac{1}{u} \, du\) to get \(\ln|u| + C\), then substitute back \(u = 6 + 3 \tan t\) to express the answer in terms of \(t\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration Using Substitution
Substitution is a method to simplify integrals by changing variables. It involves identifying a part of the integrand whose derivative also appears in the integral, allowing the integral to be rewritten in terms of a new variable. This technique is especially useful when the integral contains composite functions like trigonometric expressions.
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Substitution With an Extra Variable
Trigonometric Identities and Derivatives
Understanding the derivatives and identities of trigonometric functions such as tan(t) and sec²(t) is crucial. For example, the derivative of tan(t) is sec²(t), which helps in recognizing substitution candidates. Familiarity with these relationships simplifies the integration process involving trigonometric functions.
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Simplifying Rational Expressions in Integrals
When integrating rational expressions, simplifying the numerator and denominator can make the integral more manageable. Factoring constants and rewriting expressions help in identifying substitution opportunities and reducing the integral to a standard form that is easier to evaluate.
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6:36Simplifying Trig Expressions
Related Practice
Textbook Question
73. Find the area between the curves y=ln(x) and y=ln(2x) from x=1 to x=5.
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Textbook Question
128. Derive the formula dy/dx = 1/(1+x²) for the derivative of y = arctan(x) by differentiating both sides of the equivalent equation tan(y)=x.
Textbook Question
In Exercises 73 and 74, repeat the steps above to solve for the functions y=f(x) and x=f^(-1)(y) defined implicitly by the given equations over the interval.
73. y^(1/3) - 1 = (x+2)³, -5 ≤ x ≤ 5, x_0 = -3/2
Textbook Question
Evaluate the integrals in Exercises 53–76.
73. ∫(from 0 to ln√3) e^x dx/(1+e^(2x))
Textbook Question
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
39. y=arctan√(x²-1) + arccsc(x), x>1