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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