Indefinite Integration
Logarithmic Functions
Grade 12
Question:
<p>Evaluate: \(I = \int \left(\frac{\ln x - 1}{(\ln x)^2 + 1}\right)^2 dx\)</p>
<p>(a) \(\frac{\ln x}{x^2 + 1} + C\)</p>
<p>(b) \(\frac{\ln x}{(\ln x)^2 + 1} + C\)</p>
<p>(c) \(\frac{x}{1 + (\ln x)^2} + C\)</p>
<p>(d) \(e^x \left(\frac{x}{x^2 + 1}\right) + C\)</p>
Step-by-Step Solution
Key Concept: Substitution with logarithmic functions often simplifies the integrand to standard forms.
<p>Use substitution $u = \ln x$, $du = \frac{1}{x}dx$. The integrand simplifies to an arctangent-related form or recognizes as derivative of a composite function.</p>
Correct Answer: B