Evaluate the integrals in Exercises 31–78.
45. ∫(ln x)^(-3)/x dx

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Identify the integral to solve: \(\int \frac{(\ln x)^{-3}}{x} \, dx\).

Recognize that the integral involves a composite function with \(\ln x\) inside and a factor of \(\frac{1}{x}\) outside, which suggests using substitution.

Let \(u = \ln x\). Then, compute the differential: \(du = \frac{1}{x} \, dx\), which means \(dx = x \, du\) but since \(du = \frac{1}{x} dx\), we can rewrite \(\frac{1}{x} dx\) as \(du\).

Rewrite the integral in terms of \(u\): \(\int u^{-3} \, du\).

Integrate \(u^{-3}\) using the power rule for integrals: \(\int u^{n} \, du = \frac{u^{n+1}}{n+1} + C\) for \(n \neq -1\). Here, \(n = -3\), so the integral becomes \(\frac{u^{-2}}{-2} + C\).

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Key Concepts

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

Integration by Substitution

Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable, differentiating it, and rewriting the integral in terms of this variable to make it easier to solve.

Properties of Logarithmic Functions

Understanding the natural logarithm function, ln(x), is essential, including its domain (x > 0) and derivative (1/x). Recognizing how powers of ln(x) behave helps in manipulating and integrating expressions involving logarithms.

Integration of Power Functions

Integrating functions of the form u^n, where n is a real number, follows the power rule for integration. This rule states that ∫u^n du = u^(n+1)/(n+1) + C, provided n ≠ -1, which is crucial when integrating expressions like (ln x)^(-3).

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