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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.7.73b
Chapter 7, Problem 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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1
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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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 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
Related Practice
Textbook Question

Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.

7. a. sec^(-1)(-√2)

Textbook Question

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:


b. Solve the equation y=f(x) for x as a function of y, and name the resulting inverse function g.


70. y= x³/(x²+1), -1 ≤ x ≤ 1, x_0=1/2

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Textbook Question

Find the volumes of the solids in Exercises 135 and 136.

135. The solid lies between planes perpendicular to the x-axis at x=-1 and x=1. The cross-sections perpendicular to the x-axis are

a. circles whose diameters stretch from the curve y=-1/√(1+x²) to the curve y=1/√(1+x²).

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Textbook Question

Evaluate the integrals in Exercises 67–74 in terms of

b. natural logarithms.

67. ∫(from 0 to 2√3)dx/√(4+x²)

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Textbook Question

21. a. Show that ln(x) grows slower as x→∞ than x^(1/n) for any positive integer n, even x^(1/1,000,000).

Textbook Question

In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.

2. y' = y²

b. y = -1/(x+3)