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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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.

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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.

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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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