Evaluate : $\int \frac{\sin(x-\alpha)}{\sin(x+\alpha)}\,dx$

$(x+\alpha)\cos 2\alpha-\sin 2\alpha\,\log|\sin(x+\alpha)|+C$

CBSE Class 12 · 2006 · Indefinite IntegralsVery Short Answer

Evaluate :

\int \frac{s i n ( x - α )}{s i n ( x + α )} d x

Answer: $(x + α) cos 2 α - sin 2 α lo g ∣ sin (x + α) ∣ + C$

Put

t = x + α

, so

d x = d t

and

x - α = t - 2 α

. Expanding the numerator,

\frac{sin ( t - 2 α )}{sin t} = \frac{sin t cos 2 α - cos t sin 2 α}{sin t} = cos 2 α - sin 2 α cot t .

Integrating term by term,

\int (cos 2 α - sin 2 α cot t) d t = t cos 2 α - sin 2 α lo g ∣ sin t ∣ + C .

Restoring

t = x + α

(x + α) cos 2 α - sin 2 α lo g ∣ sin (x + α) ∣ + C

CBSE Class 12 Board · 2006 · Subjective