Class 12 Integrals · Long Answer3 Marksindefinite_integration

Class 12 Maths · Integrals · Long Answer #73

Evaluate \(\int - \frac{\sin{\left(a - x \right)}}{\sin{\left(a + x \right)}}\,dx\). Work on intervals with sin(x+a)≠0, for real constant a.

✅ Step-by-Step Solution

Final Answer: \(x \cos{\left(2 a \right)} - \log{\left(\left|{\sin{\left(a + x \right)}}\right| \right)} \sin{\left(2 a \right)}\) + C

Write sin(x−a)=sin[(x+a)−2a]=sin(x+a)cos2a−cos(x+a)sin2a.
Divide by sin(x+a) to obtain cos2a−sin2a cot(x+a).
Integrate the constant cos2a and use ∫cot(x+a)dx=ln|sin(x+a)|.
Therefore the required integral is \(x \cos{\left(2 a \right)} - \log{\left(\left|{\sin{\left(a + x \right)}}\right| \right)} \sin{\left(2 a \right)}\) + C.

[0.5 marks] Write sin(x−a)=sin[(x+a)−2a]=sin(x+a)cos2a−cos(x+a)sin2a.
[1 mark] Divide by sin(x+a) to obtain cos2a−sin2a cot(x+a).
[1 mark] Integrate the constant cos2a and use ∫cot(x+a)dx=ln|sin(x+a)|.
[0.5 marks] Therefore the required integral is \(x \cos{\left(2 a \right)} - \log{\left(\left|{\sin{\left(a + x \right)}}\right| \right)} \sin{\left(2 a \right)}\) + C.

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