Indefinite Integration
Logarithmic Functions
Grade 12
Question:
<p>Evaluate \(\int \frac{\ln x}{x(1 + \ln x)} dx\)</p>
<p>(A) \(\frac{1}{3}(1 + \ln x)^{3/2} - \frac{1}{4}\sqrt{\ln x} + C\)</p>
<p>(B) \(\frac{2}{3}(1 + \ln x)^{3/2} - \frac{1}{4}\sqrt{\ln x} + C\)</p>
<p>(C) \(\frac{2}{3}(1 + \ln x)^{3/2} - 2\sqrt{1 + \ln x} + C\)</p>
<p>(D) none of these</p>
Step-by-Step Solution
Key Concept: Use the substitution $u = 1 + \ln x$ to convert a logarithmic integral into a rational form.
<p><strong>Step 1:</strong> Let $u = 1 + \ln x$, then $du = \frac{1}{x}dx$ and $\ln x = u - 1$.</p><p><strong>Step 2:</strong> $\int \frac{\ln x}{x(1 + \ln x)} dx = \int \frac{u-1}{u} du = \int \left(1 - \frac{1}{u}\right) du$</p><p><strong>Step 3:</strong> $= u - \ln|u| + C = (1 + \ln x) - \ln(1 + \ln x) + C$</p>
Correct Answer: C