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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.4.44
Chapter 8, Problem 8.4.44

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ √(1 - (ln x)²) / (x ln x) dx

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Identify the integral: \(\int \frac{\sqrt{1 - (\ln x)^2}}{x \ln x} \, dx\).
Use the substitution \(u = \ln x\). Then, compute \(du = \frac{1}{x} dx\), which implies \(dx = x \, du\).
Rewrite the integral in terms of \(u\): substitute \(\ln x\) with \(u\) and \(dx\) with \(x \, du\). The integral becomes \(\int \frac{\sqrt{1 - u^2}}{x u} \cdot x \, du = \int \frac{\sqrt{1 - u^2}}{u} \, du\).
Now, to handle the integral \(\int \frac{\sqrt{1 - u^2}}{u} \, du\), use a trigonometric substitution. Since the integrand contains \(\sqrt{1 - u^2}\), let \(u = \sin \theta\), which implies \(du = \cos \theta \, d\theta\).
Rewrite the integral in terms of \(\theta\): substitute \(u = \sin \theta\) and \(du = \cos \theta \, d\theta\). The integral becomes \(\int \frac{\sqrt{1 - \sin^2 \theta}}{\sin \theta} \cdot \cos \theta \, d\theta\). Simplify the square root using the Pythagorean identity and proceed to integrate with respect to \(\theta\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Substitution Method in Integration

The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. It often involves identifying a part of the integrand whose derivative also appears, allowing the integral to be rewritten in terms of a new variable.
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Euler's Method

Trigonometric Substitution

Trigonometric substitution replaces algebraic expressions involving square roots with trigonometric functions to simplify integration. It is especially useful for integrals containing expressions like √(1 - u²), where substituting u = sin θ or u = cos θ converts the integral into a trigonometric form.
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Introduction to Trigonometric Functions

Integration of Functions Involving Logarithms

Integrals involving logarithmic functions often require careful substitution since the logarithm's derivative is 1/x. Recognizing when to substitute variables like ln x can simplify the integral, especially when combined with other techniques such as trigonometric substitution.
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Integrals Involving Natural Logs: Substitution
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