Indefinite Integration
Integration with Functional Relations
Grade 12
Question:
<p>If \(xf(x) = 3f^2(x) + 2\), then \(\int \frac{2x - 12xf(x) - f(x)}{(6f(x) - x)(x^2 - f(x))^2} dx\) equals</p>
<p>(A) \(\frac{1}{2(x - f(x))^2} + C\)</p>
<p>(B) \(\frac{1}{2(x + f(x))^2} + C\)</p>
<p>(C) \(\frac{1}{(x - f(x))^2} + C\)</p>
<p>(D) \(\frac{1}{(x + f(x))^2} + C\)</p>
Step-by-Step Solution
Key Concept: Use the given constraint to simplify the integrand and recognize it as a derivative of a composite expression.
<p>Use the constraint $xf(x) = 3f^2(x) + 2$ to simplify the integrand.</p><p>Recognize that the numerator is the derivative of some composite function involving $(x - f(x))$.</p><p>Evaluate the integral using substitution: Let $u = x - f(x)$, then $\int \frac{du}{u^2} = -\frac{1}{u} + C = -\frac{1}{x - f(x)} + C$</p><p>∴ Answer is A.</p>
Correct Answer: A