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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
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Tutti i libri di testoHass 15th EditionCh. 7 - Transcendental FunctionsProblema 7.7.73b
Capitolo 7, Problema 7.7.73b

Evaluate the integrals in Exercises 67–74 in terms of
b. natural logarithms.
73. ∫(from 0 to π)cos(x)dx/√(1+sin²x)

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Identify the integral to evaluate: \(\displaystyle \int_0^{\pi} \frac{\cos(x)}{\sqrt{1 + \sin^2(x)}} \, dx\).
Recognize that the integrand contains \(\cos(x)\) and \(\sin(x)\), suggesting a substitution involving \(\sin(x)\) might simplify the integral.
Use the substitution \(u = \sin(x)\), which implies \(du = \cos(x) \, dx\). This transforms the integral into \(\int_{u=\sin(0)}^{u=\sin(\pi)} \frac{1}{\sqrt{1 + u^2}} \, du\).
Evaluate the new limits: since \(\sin(0) = 0\) and \(\sin(\pi) = 0\), the integral becomes \(\int_0^0 \frac{1}{\sqrt{1 + u^2}} \, du\).
Notice that the limits are the same, so the integral evaluates to zero. However, consider the behavior of the function on the interval and whether the substitution covers the entire integral correctly, possibly splitting the integral at \(\pi/2\) to handle the sign of \(\cos(x)\).

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Definite Integrals

A definite integral calculates the net area under a curve between two specific limits. It is represented as ∫ from a to b of f(x) dx, where a and b are the bounds. Evaluating definite integrals often involves finding an antiderivative and then applying the Fundamental Theorem of Calculus.
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Definition of the Definite Integral

Substitution Method

The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. It is especially useful when the integrand contains a composite function, allowing the integral to be expressed in terms of a new variable, often leading to easier integration.
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Euler's Method

Natural Logarithms in Integration

Natural logarithms often appear when integrating functions of the form f'(x)/f(x), resulting in ln|f(x)| + C. Recognizing when an integral can be expressed in terms of natural logarithms helps simplify the evaluation, especially when the integrand involves expressions under a root or rational functions.
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Derivative of the Natural Logarithmic Function
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Evaluate the integrals in Exercises 67–74 in terms of

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In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:


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