Use any method to evaluate the integrals in Exercises 55–66.
∫ 2 / (x(ln x - 2)³) dx

Verified step by step guidance

1. Identify the integral to solve: \(\int \frac{2}{x(\ln x - 2)^3} \, dx\).
2. Recognize that the integral involves a composite function \(\ln x - 2\) in the denominator, suggesting a substitution related to \(\ln x\).
3. Let \(u = \ln x - 2\). Then, compute the differential \(du\): since \(\frac{d}{dx}(\ln x) = \frac{1}{x}\), we have \(du = \frac{1}{x} dx\).
4. Rewrite the integral in terms of \(u\) and \(du\). Notice that \(dx = x \, du\), so substitute accordingly to express the integral fully in terms of \(u\).
5. After substitution, the integral becomes \(\int \frac{2}{u^3} \, du\). Now, integrate this expression using the power rule for integrals.

Key Concepts

Integration by Substitution

Integration by substitution is a technique used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable, which transforms the integral into a simpler form. This method is especially useful when the integral contains a composite function, such as ln(x) in this problem.

Properties of Logarithmic Functions

Understanding the properties of logarithmic functions, like ln(x), is essential for manipulating and simplifying expressions involving logs. Recognizing how derivatives of ln(x) relate to 1/x helps in choosing appropriate substitutions and solving integrals involving logarithmic terms.

Handling Rational Functions in Integration

Rational functions are ratios of polynomials or expressions involving variables in numerator and denominator. Integrating rational functions often requires algebraic manipulation or substitution to simplify the integrand. In this problem, the presence of x in the denominator combined with a function of ln(x) suggests careful handling of the rational expression.