<p>Evaluate <span class="math">\int_{50}^{100} \frac{\ln x}{\ln x + \ln(150-x)} dx</span></p>
Step-by-Step Solution
Key Concept: Use the substitution property of definite integrals where replacing x with (a-x) helps simplify symmetric integrands. Adding the original and transformed integrals eliminates the complex denominator.
<p><strong>Step 1:</strong> Let <span class="math">I = \int_{50}^{100} \frac{\ln x}{\ln x + \ln(150-x)} dx</span></p><p><strong>Step 2:</strong> Using property P-5, replace <span class="math">x</span> with <span class="math">150-x</span>:</p><p><span class="math">I = \int_{50}^{100} \frac{\ln(150-x)}{\ln(150-x) + \ln x} dx</span></p><p><strong>Step 3:</strong> Add the two expressions:</p><p><span class="math">2I = \int_{50}^{100} \frac{\ln x + \ln(150-x)}{\ln x + \ln(150-x)} dx = \int_{50}^{100} 1 \, dx = 50</span></p><p><strong>Step 4:</strong> Therefore, <span class="math">I = 25</span></p>
Correct Answer: 25