Evaluate the integrals in Exercises 39–56.
47. ∫(from 2 to 4)dx/(x(ln x)²)

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Identify the integral to be evaluated: \(\int_{2}^{4} \frac{dx}{x (\ln x)^2}\).

Recognize that the integrand involves \(x\) and \(\ln x\), suggesting a substitution related to \(\ln x\).

Let \(u = \ln x\). Then, compute \(du\): since \(u = \ln x\), we have \(du = \frac{1}{x} dx\), which implies \(dx = x \, du\).

Rewrite the integral in terms of \(u\): substitute \(dx\) and \(x\) accordingly. The integral becomes \(\int_{u=\ln 2}^{u=\ln 4} \frac{x \, du}{x u^2} = \int_{\ln 2}^{\ln 4} \frac{du}{u^2}\).

Now, integrate \(\int \frac{1}{u^2} du = \int u^{-2} du\). Use the power rule for integration to find the antiderivative, then apply the limits \(u=\ln 2\) to \(u=\ln 4\).

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Key Concepts

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

Integration of Rational Functions

This involves integrating functions expressed as ratios of polynomials or other expressions. Recognizing the form helps in choosing appropriate techniques such as substitution or partial fractions to simplify the integral.

Substitution Method

A technique where a part of the integral is replaced with a new variable to simplify the integral. For example, substituting u = ln(x) transforms the integral into a more manageable form involving u.

Properties of the Natural Logarithm

Understanding the behavior and derivatives of ln(x) is crucial. Since d/dx(ln x) = 1/x, this relationship often guides substitution choices and helps simplify integrals involving ln(x).

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