Evaluate the integrals in Exercises 97–110.
107. ∫₀⁹ (2 log₁₀(x + 1) / (x + 1)) dx

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Recognize that the integral is \( \int_0^9 \frac{2 \log_{10}(x+1)}{x+1} \, dx \). The presence of \( \log_{10}(x+1) \) divided by \( x+1 \) suggests a substitution related to the logarithm's argument.

Recall the change of base formula for logarithms: \( \log_{10}(x+1) = \frac{\ln(x+1)}{\ln(10)} \). Rewrite the integral using natural logarithms to simplify differentiation and integration.

Substitute \( u = \ln(x+1) \). Then, compute \( du = \frac{1}{x+1} dx \), which matches part of the integrand, allowing us to rewrite the integral in terms of \( u \).

Rewrite the integral as \( \int_{u=\ln(1)}^{u=\ln(10)} 2 \cdot \frac{u}{\ln(10)} \, du \) by substituting the limits accordingly and simplifying the integrand.

Integrate the resulting expression with respect to \( u \), then substitute back if necessary, and finally evaluate the definite integral using the new limits.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Integration of Logarithmic Functions

Integrating functions involving logarithms often requires substitution or recognizing derivative patterns. For example, the integral of (log(x))/x can be approached by substitution or integration by parts, leveraging the relationship between logarithms and their derivatives.

Change of Variable (Substitution Method)

Substitution simplifies integrals by changing variables to transform the integral into a more familiar or easier form. In this problem, substituting u = x + 1 can simplify the integral limits and the integrand, making the logarithmic expression easier to handle.

Properties of Logarithms with Different Bases

Logarithms with bases other than e can be converted using the change of base formula: log_a(b) = ln(b)/ln(a). Understanding this allows rewriting log base 10 in terms of natural logarithms, which are more convenient for integration and differentiation.

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