Indefinite Integration
Integration of Trigonometric Functions
Grade 12
Question:
<p>\(\displaystyle\int\frac{\sin(x-a)}{\sin(x+a)}\,dx\) equals</p>
<li>\(x\cos 2a - \sin 2a\cdot\ln|\sin(x+a)|+C\)</li>
<li>\(x\cos 2a + \sin 2a\cdot\ln|\sin(x+a)|+C\)</li>
<li>\(x\sin 2a - \cos 2a\cdot\ln|\sin(x+a)|+C\)</li>
<li>\(-x\cos 2a + \sin 2a\cdot\ln|\cos(x+a)|+C\)</li>
Step-by-Step Solution
Key Concept: Write sin(x-a) = sin((x+a)-2a) = sin(x+a)cos2a - cos(x+a)sin2a. Divide to get cos2a - sin2a \cdot cot(x+a).
<p><strong>Expand numerator:</strong> $\sin(x-a)=\sin((x+a)-2a)=\sin(x+a)\cos 2a-\cos(x+a)\sin 2a$.</p>
<p>$$\frac{\sin(x-a)}{\sin(x+a)} = \cos 2a - \sin 2a\cdot\cot(x+a)$$</p>
<p>$$\int\left(\cos 2a - \sin 2a\cdot\cot(x+a)\right)dx = x\cos 2a - \sin 2a\cdot\ln|\sin(x+a)|+C$$</p>
<p>Answer: <strong>(A)</strong></p>
Correct Answer: A