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

검증된 단계별 안내

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

Rewrite the integral using the natural logarithm: \( \int \frac{\log_{10} x}{x} \, dx = \int \frac{\frac{\ln x}{\ln 10}}{x} \, dx = \frac{1}{\ln 10} \int \frac{\ln x}{x} \, dx \).

Focus on the integral \( \int \frac{\ln x}{x} \, dx \). To solve this, use substitution: let \( t = \ln x \), which implies \( dt = \frac{1}{x} dx \).

Rewrite the integral in terms of \( t \): \( \int t \, dt \), since \( \frac{\ln x}{x} dx = t \, dt \).

Integrate \( \int t \, dt \) to get \( \frac{t^2}{2} + C \), then substitute back \( t = \ln x \) to express the answer in terms of \( x \). Finally, multiply by the constant \( \frac{1}{\ln 10} \) from step 2.

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Integration of Logarithmic Functions

Integrating logarithmic functions often involves recognizing the form and applying substitution or integration by parts. For example, integrals involving log(x) can be simplified by rewriting the logarithm or using properties of logarithms to facilitate integration.

Change of Logarithm Base

Logarithms with different bases can be converted using the formula log_a(x) = ln(x) / ln(a). This is useful because natural logarithms (ln) are easier to integrate and differentiate, allowing the integral to be expressed in terms of ln(x) for simpler calculation.

Substitution Method in Integration

The substitution method involves changing variables to simplify an integral. For integrals like ∫ (log₁₀x / x) dx, setting u = log₁₀x or u = ln(x) can transform the integral into a more manageable form, making it easier to evaluate.

Evaluate the integrals in Exercises 97–110.101. ∫ (log₁₀x / x) dx