Evaluate the integrals in Exercises 97–110.
103. ∫₁⁴ (ln 2 · log₂x / x) dx

Verified step by step guidance

1. Recognize that the integral is \( \int_1^4 \frac{\ln 2 \cdot \log_2 x}{x} \, dx \). Notice that \( \ln 2 \) is a constant and \( \log_2 x \) is the logarithm base 2 of \( x \).
2. Recall the change of base formula for logarithms: \( \log_2 x = \frac{\ln x}{\ln 2} \). Substitute this into the integral to rewrite the integrand in terms of natural logarithms.
3. After substitution, the integrand becomes \( \frac{\ln 2 \cdot \frac{\ln x}{\ln 2}}{x} = \frac{\ln x}{x} \). This simplifies the integral to \( \int_1^4 \frac{\ln x}{x} \, dx \).
4. To solve \( \int \frac{\ln x}{x} \, dx \), use the substitution \( t = \ln x \), which implies \( dt = \frac{1}{x} dx \). This transforms the integral into \( \int t \, dt \).
5. Integrate \( \int t \, dt \) to get \( \frac{t^2}{2} + C \). Substitute back \( t = \ln x \) to express the antiderivative as \( \frac{(\ln x)^2}{2} + C \). Finally, evaluate this expression at the limits \( x=1 \) and \( x=4 \) to find the definite integral.

Key Concepts

Change of Logarithm Base

The integral involves logarithms with different bases (natural log and base 2). Converting all logarithms to a common base, typically the natural logarithm, simplifies the expression and makes integration manageable. Use the formula log_a(x) = ln(x) / ln(a) to rewrite log₂(x) in terms of ln(x).

Properties of Logarithms

Understanding logarithm properties, such as the product, quotient, and power rules, helps simplify the integrand. In this problem, recognizing that ln(2)·log₂(x) can be expressed as ln(x) is key to reducing the integral to a simpler form that is easier to integrate.

Integration of Logarithmic Functions

Integrating functions involving logarithms often requires techniques like substitution or integration by parts. For example, integrating (ln x)/x dx can be done by recognizing it as the derivative of (ln x)^2 / 2. Familiarity with these methods is essential to evaluate the integral correctly.