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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 8 - Techniques of IntegrationProblema 8.2.34
Capitolo 8, Problema 8.2.34

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

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

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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.
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Substitution With an Extra Variable

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.
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Properties of Functions

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.
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Integration by Parts for Definite Integrals
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