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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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.

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.

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.

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