Q101 / 180 · KCET 2024 · Mathematics
$\int \frac{\sin \frac{5 x}{2}}{\sin \frac{x}{2}} d x$ is
- A$2 x+\sin x+2 \sin 2 x+C$
- B$x+2 \sin x+2 \sin 2 x+C$
- Correct answerC$x+2 \sin x+\sin 2 x+C$
- D$2 x+\sin x+\sin 2 x+C$
Solution
$I=\int \frac{\sin \left(\frac{5 x}{2}\right)}{\sin \left(\frac{x}{2}\right)} d x$
$=\int \frac{2 \sin \left(\frac{5 x}{2}\right) \cos \left(\frac{x}{2}\right)}{\sin x} d x=\int \frac{\sin 3 x+\sin 2 x}{\sin x} d x$
$=\int \frac{\left(3 \sin x-4 \sin ^3 x\right)+2 \sin x \cos x}{\sin x} d x$
$=\int\left(3-4 \sin ^2 x+2 \cos x\right) d x$
$=\int(3-2(1-\cos 2 x)+2 \cos x) d x$
$=\int(3-2+2 \cos 2 x+2 \cos x) d x$
$=\int(1+2 \cos 2 x+2 \cos x) d x$
$=x+\sin 2 x+2 \sin x+C$
Hence, the correct answer is option: 'C'.