Evaluate the integrals in Exercises 39–56.
45. ∫(from 1 to 2)(2ln x)/x dx

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Identify the integral to be evaluated: \(\int_{1}^{2} \frac{2 \ln x}{x} \, dx\).

Recognize that the integrand involves \(\ln x\) divided by \(x\), which suggests a substitution related to \(\ln x\).

Let \(u = \ln x\). Then, compute the differential \(du = \frac{1}{x} dx\), which implies \(dx = x \, du\).

Rewrite the integral in terms of \(u\): since \(\frac{2 \ln x}{x} dx = 2u \cdot \frac{1}{x} dx = 2u \, du\).

Change the limits of integration according to \(u = \ln x\): when \(x=1\), \(u=\ln 1=0\); when \(x=2\), \(u=\ln 2\). Then, the integral becomes \(\int_{0}^{\ln 2} 2u \, du\), which can be integrated using the power rule.

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Key Concepts

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

Integration of Logarithmic Functions

Integrating functions involving logarithms often requires recognizing patterns or using substitution. For example, integrals of the form ∫(ln x)/x dx can be simplified by substitution or by recalling standard integral results involving logarithms.

Definite Integrals and Limits of Integration

Definite integrals compute the net area under a curve between two points, using specified limits of integration. Evaluating a definite integral involves finding the antiderivative and then applying the Fundamental Theorem of Calculus to substitute the upper and lower limits.

Properties of Natural Logarithm and Its Derivative

The natural logarithm function ln(x) has a derivative of 1/x, which is useful in integration techniques. Understanding how ln(x) behaves and its relationship with its derivative helps in simplifying integrals involving ln(x) and x in the denominator.

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f(x) = x² − 2x, x ≤ 1

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