NCERT Class 12 · Ch 7 Miscellaneous Exercise · Q7
NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 7
Evaluate \(\int \left(- \frac{\sin{\left(x \right)}}{\sin{\left(a - x \right)}}\right)\, dx\).
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: \(x \cos{\left(a \right)} + \log{\left(\left|{\sin{\left(a - x \right)}}\right| \right)} \sin{\left(a \right)}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Expand sin x=sin(x−a)cos a+cos(x−a)sin a; divide and integrate cot(x−a).
- An antiderivative is F(x)=\(x \cos{\left(a \right)} + \log{\left(\left|{\sin{\left(a - x \right)}}\right| \right)} \sin{\left(a \right)}\).
- Differentiate this expression to check that F′(x)=\(- \frac{\sin{\left(x \right)}}{\sin{\left(a - x \right)}}\). Add an arbitrary constant C on each domain component.
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