Indefinite Integration
Integration using Trigonometric Identities
Grade 12
Question:
<p>If <span style='display:inline-block'>∫</span> \(\frac{\cos 4x}{\sin^2 x}\) dx = \(A \cot x + B \sin 2x + C\), then find A and B.</p>
<p>(a) \(A = -2, B = \frac{1}{4}\)</p>
<p>(b) \(B = -\frac{1}{4}, C = -3\)</p>
<p>(c) \(B = \frac{1}{4}, C = -3\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Decompose the integrand using trigonometric identities and integrate term by term.
<p><strong>Solution:</strong></p><p>We have: $\int \frac{\cos 4x}{\sin^2 x}$ dx = $\int \frac{\sin^2 x - (1-\sin^2 x)^2}{\sin^2 x}$ dx</p><p>= $\int (\cosec^2 x + \sin 2x - 2)$ dx</p><p>= $-\cot x - \frac{1}{2}\cos 2x - 2x + C$</p><p>Comparing with $A \cot x + B \sin 2x + C$: $A = -1, B = -\frac{1}{4}$</p><p>∴ Answer is (b).</p>
Correct Answer: B