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

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Integration of Rational Functions

This involves integrating functions expressed as ratios of polynomials or other expressions. Recognizing the form of the integrand helps in choosing appropriate substitution or integration techniques.

Logarithmic Functions and Their Properties

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.

Substitution Method in Integration

This technique involves substituting a part of the integrand with a new variable to simplify the integral. For integrals involving logarithms, substituting the logarithmic expression often reduces the integral to a basic form.

Evaluate the integrals in Exercises 97–110.109. ∫ (dx / (x log₁₀x))