Textbook Question
Evaluate the integrals in Exercises 31–78.
31. ∫e^x sin(e^x)dx
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.61
Evaluate the integrals in Exercises 31–78.
61. ∫(from 1 to 3)(ln(v+1))²/(v+1) dv
Verified step by step guidance1
Recognize that the integral is of the form \( \int_{1}^{3} \frac{(\ln(v+1))^2}{v+1} \, dv \). To simplify, use the substitution \( u = v + 1 \).
Compute the differential: since \( u = v + 1 \), then \( du = dv \). Also, change the limits of integration accordingly: when \( v = 1 \), \( u = 2 \); when \( v = 3 \), \( u = 4 \).
Rewrite the integral in terms of \( u \): \( \int_{2}^{4} \frac{(\ln u)^2}{u} \, du \).
Recognize that this integral can be approached by using the substitution \( t = \ln u \), which implies \( dt = \frac{1}{u} du \). This transforms the integral into \( \int_{t=\ln 2}^{t=\ln 4} t^2 \, dt \).
Integrate \( t^2 \) with respect to \( t \) over the new limits, then substitute back if needed to express the answer in terms of \( v \).
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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 substituting a part of the integrand with a new variable to transform the integral into a more manageable form. For example, setting u = v + 1 can simplify the given integral by reducing the complexity of the integrand.
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Properties of Logarithmic Functions
Understanding logarithmic functions, especially natural logarithms (ln), is essential. The function ln(x) is the inverse of the exponential function e^x, and its properties, such as the chain rule in differentiation and integration, help in manipulating integrals involving ln expressions.
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Definite Integrals and Limits of Integration
Definite integrals calculate the net area under a curve between two points. Evaluating a definite integral requires applying the Fundamental Theorem of Calculus, which involves finding an antiderivative and then computing the difference of its values at the upper and lower limits.
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Related Practice
Textbook Question
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