Indefinite Integration
Special Integration Formulas
Grade 12
Question:
<p>The value of \(\int \frac{f(x)\phi'(x) - \phi(x)f'(x)}{[f(x)\phi(x)-1]^2 - f(x)\phi(x) - 1} dx\) is</p>
<p>(A) \(\sin^{-1}\left(\frac{f(x)}{\phi(x)}\right)\)</p>
<p>(B) \(\cos^{-1}\left(\frac{f(x)}{\sqrt{f^2(x) + \phi^2(x)}}\right)\)</p>
<p>(C) \(\tan^{-1}[f(x) \cdot \phi(x)]\)</p>
<p>(D) None of these</p>
Step-by-Step Solution
Key Concept: Recognize the numerator as a derivative of a quotient and use inverse trigonometric integral formulas.
<p>The numerator is the derivative of the quotient $\frac{f(x)}{\phi(x)}$. The integral simplifies to an inverse sine function using standard formulas.</p>
Correct Answer: A