\int \frac{\sin \frac{5x}{2}}{\sin \frac{x}{2}}dx is equal to :

(Where c is a constant of integration.)
 

  • Option 1)

    2x+\sin x+\sin 2x+c

  • Option 2)

    2x+\sin x+2\sin 2x+c

  • Option 3)

    x+2\sin x+\sin 2x+c

     

  • Option 4)

    x+2\sin x+2\sin 2x+c

 

Answers (1)

\int \frac{\sin \frac{5x}{2}}{\sin \frac{x}{2}}dx

=\int \frac{\sin (2x+\frac{x}{2})}{\sin \frac{x}{2}}\; dx=\int \frac{\sin 2x\: \cos \frac{x}{2}+\cos 2x\: \sin \frac{x}{2}}{\sin \frac{x}{2}}\; dx

=\int \frac{2\sin x\cos x\cos \frac{x}{2}+\cos 2x\: \sin \frac{x}{2}}{\sin \frac{x}{2}}

=\int \left ( 4\cos \frac{x}{2}.\cos x.\cos \frac{x}{2}+\cos 2x \right )\; dx

=\int \left ( 4\: \cos^{2}\frac{x}{2}\cos x +\cos 2x\right )\: dx

=\int \left ( 2\left ( 1+\cos x \right )\cos x+\cos 2x \right )\: dx

=\int (2\cos x+2\cos 2x+1)\: dx

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


Option 1)

2x+\sin x+\sin 2x+c

Option 2)

2x+\sin x+2\sin 2x+c

Option 3)

x+2\sin x+\sin 2x+c

 

Option 4)

x+2\sin x+2\sin 2x+c

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