A) \[x+2\sin x+\sin 2x+C\]
B) \[x+2\cos x+\sin 2x+C\]
C) \[x-2\sin x+\sin 2x+C\]
D) \[x+2\sin x-\sin 2x+C\]
Correct Answer: A
Solution :
| \[\int{\frac{\sin \frac{5x}{2}}{\sin \frac{x}{2}}}dx=\int{\frac{2\sin \frac{5x}{2}\cos \frac{x}{2}}{2\sin \frac{x}{2}\cos \frac{x}{2}}}dx\] |
| \[=\int{\left( \frac{\sin 3x+\sin 2x}{\sin x} \right)}dx\] |
| \[=\int{2\cos xdx+\int{(3-4si{{n}^{2}}x)}}dx\] |
| \[=2\int{\cos xdx+\int{dx+2}}\int{\cos 2xdx}\] |
| \[=2\sin x+x+\sin 2x+C.\] |
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