Indefinite Integration
Trigonometric Integration
Grade None
Question:
<p>If \(\int \frac{\sin x}{\sin(x-a)} dx = Ax + B \log \sin(x-a) + C\), then the value of \((A, B)\) is:</p>
<p>(a) \((\sin a, \cos a)\)</p>
<p>(b) \((\cos a, \sin a)\)</p>
<p>(c) \((-\sin a, \cos a)\)</p>
<p>(d) \((-\cos a, \sin a)\)</p>
Step-by-Step Solution
Key Concept: Differentiate the proposed antiderivative and match it with the original integrand using trigonometric identities.
<p>Differentiate the right side and compare with the integrand. Using the product rule and chain rule on $Ax + B\log\sin(x-a)$ yields $A + B\cot(x-a)$. Rewrite $\sin x = \sin(x-a+a) = \sin(x-a)\cos a + \cos(x-a)\sin a$, then divide by $\sin(x-a)$ to get $\cos a + \cot(x-a)\sin a$. Comparing coefficients: $A = -\cos a$ and $B = \sin a$.</p>
Correct Answer: D