Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ 1/(x(ln(x))²) dx

Verified step by step guidance

1. Identify the integral to solve: \(\int \frac{1}{x (\ln(x))^{2}} \, dx\).
2. Recognize that this integral suggests a substitution because of the composite function \(\ln(x)\) inside the denominator.
3. Let \(u = \ln(x)\). Then, compute the differential: \(du = \frac{1}{x} \, dx\), which implies \(dx = x \, du\).
4. Rewrite the integral in terms of \(u\): substitute \(\ln(x)\) with \(u\) and \(\frac{1}{x} dx\) with \(du\), so the integral becomes \(\int \frac{1}{u^{2}} \, du\).
5. Integrate \(\int u^{-2} \, du\) using the power rule for integrals, then substitute back \(u = \ln(x)\) to express the answer in terms of \(x\).

Key Concepts

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, which transforms the integral into a simpler form. For example, substituting u = ln(x) can simplify integrals involving logarithmic functions.

Properties of Logarithmic Functions

Understanding logarithmic functions, especially their derivatives and integrals, is crucial. The derivative of ln(x) is 1/x, which often helps in recognizing substitution candidates. Familiarity with how powers of ln(x) behave in integrals aids in selecting the right integration technique.

Integration Techniques Beyond Integration by Parts

Not all integrals require integration by parts; some can be solved using simpler methods like substitution or recognizing standard integral forms. Identifying when to apply these techniques efficiently saves time and simplifies the problem-solving process.