Indefinite Integration
Integration using Logarithmic Functions
Grade 12
Question:
<p>Evaluate <span class="math">\(\int \frac{f(x)g'(x) - f'(x)g(x)}{f(x)g(x)} \left[\log(g(x)) - \log(f(x))\right]\, dx\)</span></p>
<p>(a) <span class="math">\(\log\left(\frac{g(x)}{f(x)}\right) + C\)</span></p>
<p>(b) <span class="math">\(\frac{1}{2}\left[\log\left(\frac{g(x)}{f(x)}\right)\right]^2 + C\)</span></p>
<p>(c) <span class="math">\(\left[\log\left(\frac{g(x)}{f(x)}\right)\right]^2 + C\)</span></p>
<p>(d) <span class="math">\(\log\left(\left[\log\left(\frac{g(x)}{f(x)}\right)\right]^2\right) + C\)</span></p>
Step-by-Step Solution
Key Concept: Recognize the logarithmic derivative formula and use substitution to convert to a simple power integral.
<p><strong>Step 1:</strong> Recognize that <span class="math">$\frac{d}{dx}\left[\log\left(\frac{g(x)}{f(x)}\right)\right] = \frac{f(x)g'(x) - f'(x)g(x)}{f(x)g(x)}$</span></p><p><strong>Step 2:</strong> Let <span class="math">$u = \log\left(\frac{g(x)}{f(x)}\right)$</span>, then <span class="math">$du = \frac{f(x)g'(x) - f'(x)g(x)}{f(x)g(x)}\, dx$</span></p><p><strong>Step 3:</strong> The integral becomes <span class="math">$\int u\, du = \frac{u^2}{2} + C = \frac{1}{2}\left[\log\left(\frac{g(x)}{f(x)}\right)\right]^2 + C$</span></p><p>∴ Answer is B.</p>
Correct Answer: B