Textbook Question
Evaluate the integrals in Exercises 31–78.
47. ∫(1/r)csc²(1+ln(r))dr
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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.PE.43
Evaluate the integrals in Exercises 31–78.
43. ∫tan(ln v)/v dv
Verified step by step guidance1
Recognize that the integral is of the form \(\int \frac{\tan(\ln v)}{v} \, dv\). Notice the presence of \(\ln v\) inside the tangent function and the \(\frac{1}{v}\) factor outside, which suggests a substitution involving \(\ln v\).
Let \(u = \ln v\). Then, compute the differential \(du\): since \(u = \ln v\), we have \(du = \frac{1}{v} dv\). This means \(\frac{1}{v} dv = du\).
Rewrite the integral in terms of \(u\): substituting \(u\) and \(du\) gives \(\int \tan(u) \, du\).
Recall the integral formula for \(\int \tan(u) \, du\). The integral of \(\tan(u)\) is \(-\ln|\cos(u)| + C\). Use this to express the integral in terms of \(u\).
Finally, substitute back \(u = \ln v\) to write the answer in terms of the original variable \(v\). The result will be \(-\ln|\cos(\ln v)| + C\).
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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 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 integral contains a composite function, such as tan(ln v).
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Substitution With an Extra Variable
Properties of Logarithmic Functions
Logarithmic functions, like ln(v), have specific properties that help in integration, such as their derivatives and the chain rule application. Understanding how ln(v) behaves and how its derivative 1/v appears in the integral is crucial for choosing the correct substitution.
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Integration of Trigonometric Functions
Integrating trigonometric functions like tan(x) requires knowledge of their standard integrals. For example, ∫tan(x) dx = -ln|cos(x)| + C. Recognizing this allows one to integrate tan(ln v) after substitution, linking trigonometric integration with logarithmic substitution.
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Related Practice
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Textbook Question
Evaluate the integrals in Exercises 31–78.
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