Ch. 7 - Transcendental Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problema 7.3.109

Capitolo 7, Problema 7.3.109

Evaluate the integrals in Exercises 97–110.
109. ∫ (dx / (x log₁₀x))

Guida verificata passo dopo passo

Recognize that the integral involves the expression \(\frac{1}{x \log_{10} x}\). Since the logarithm is base 10, recall the change of base formula: \(\log_{10} x = \frac{\ln x}{\ln 10}\), where \(\ln\) is the natural logarithm.

Rewrite the integral using the change of base formula: \(\int \frac{dx}{x \log_{10} x} = \int \frac{dx}{x \frac{\ln x}{\ln 10}} = \int \frac{\ln 10}{x \ln x} \, dx\).

Factor out the constant \(\ln 10\) from the integral: \(\ln 10 \int \frac{1}{x \ln x} \, dx\).

Use substitution to solve the integral \(\int \frac{1}{x \ln x} \, dx\). Let \(u = \ln x\), then \(du = \frac{1}{x} dx\), which means \(dx = x \, du\).

Substitute into the integral: \(\int \frac{1}{x u} dx = \int \frac{1}{x u} (x \, du) = \int \frac{1}{u} du\). This integral is \(\ln |u| + C = \ln |\ln x| + C\). Don't forget to multiply back by the constant \(\ln 10\).

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Understanding the properties of logarithms, especially change of base formulas and derivatives, is essential. For example, log₁₀(x) can be expressed in terms of natural logarithms, which simplifies integration.

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