Indefinite Integration
Trigonometric Integrals
IIT-JEE 1983
Grade 12
Question:
$\int \frac{\sin x}{\sin(x - a)} \, dx$ equals:
(1) $x \cos a + \sin a \ln |\sin(x - a)| + C$
(2) $x \sin a - \cos a \ln |\sin(x - a)| + C$
(3) $x \cos a - \sin a \ln |\sin(x - a)| + C$
(4) $x \sin a + \cos a \ln |\sin(x - a)| + C$
Step-by-Step Solution
Key Concept: Write $\sin x = \sin[(x - a) + a] = \sin(x - a) \cos a + \cos(x - a) \sin a$ and split the integral into two simpler pieces.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)