Question

Evaluate the integrals in Exercises 97–110.105. ∫₀² (log₂(x + 2) / (x + 2)) dx

Accepted Answer

Recognize that the integral is $$ \int_0^2 \frac{\log_2(x + 2)}{x + 2} \, dx . $$The logarithm is base 2, so recall the change of base formula: $$ \log_2(y) = \frac{\ln(y)}{\ln(2)} $$, where $$ \ln  is $$the natural logarithm. Rewrite the integral using the change of base formula: $$ \int_0^2 \frac{\log_2(x + 2)}{x + 2} \, dx = \int_0^2 \frac{\frac{\ln(x + 2)}{\ln(2)}}{x + 2} \, dx = \frac{1}{\ln(2)} \int_0^2 \frac{\ln(x + 2)}{x + 2} \, dx. $$ Make the substitution $$ t = x + 2 . $$Then, $$ dt = dx $$, and when $$ x = 0 $$, $$ t = 2 $$; when $$ x = 2 $$, $$ t = 4 . $$The integral becomes $$ \frac{1}{\ln(2)} \int_2^4 \frac{\ln(t)}{t} \, dt. $$ Focus on evaluating $$ \int_2^4 \frac{\ln(t)}{t} \, dt . $$Consider the substitution $$ u = \ln(t) $$, which implies $$ du = \frac{1}{t} dt . $$This transforms the integral into $$ \int u \, du . $$Integrate $$ \int u \, du  to $$get $$ \frac{u^2}{2} + C . $$Substitute back $$ u = \ln(t)  to $$express the antiderivative as $$ \frac{(\ln(t))^2}{2} + C . $$Finally, apply the limits from 2 to 4 and multiply by $$ \frac{1}{\ln(2)}  to $$complete the evaluation.

Evaluate the integrals in Exercises 97–110.
105. ∫₀² (log₂(x + 2) / (x + 2)) dx

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Concetti chiave

Change of Base Formula for Logarithms

Substitution Method in Integration

Integral of (ln u) / u