, 5x 2 ,x 2 dx મેળવો. - Vidyadip

MCQ

$\int {\frac{{\sin \,\frac{{5x}}{2}}}{{\sin \,\frac{x}{2}}}} dx$ મેળવો.

A
$x + 2\,\sin \,x + 2\,\sin \,2x + c$
B
$2x + \,\sin \,x + 2\,\sin \,2x + c$
✓
$x + 2\,\sin \,x + \,\sin \,2x + c$
D
$2x + \,\sin \,x + \,\sin \,2x + c$

✓

Answer

Correct option: C.

$x + 2\,\sin \,x + \,\sin \,2x + c$

c
$\int \frac{\sin \frac{5 x}{2}}{\sin \frac{x}{2}} d $

$x=\int \frac{2 \sin \frac{5 x}{2} \cos \frac{x}{2}}{2 \sin \frac{x}{2} \cos \frac{x}{2}} d x$

$=\int \frac{\sin 3 x+\sin 2 x}{\sin x} d x$

$=\int \frac{3 \sin x-4 \sin ^{3} x+2 \sin x \cos x}{\sin x} d x$

$=\int\left(3-4 \sin ^{2} x+2 \cos x\right) d x$

$=\int(3-2(1-\cos 2 x)+2 \cos x) d x$

$=\int(1+2 \cos 2 x+2 \cos x) d x$

$=x+\sin 2 x+2 \sin x+c$

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