$\int \frac{\sin \frac{5 x}{2}}{\sin \frac{x}{2}} d x=$ (where $C$ is a constant of integration.)

Question

Aicharya · Accepted Answer

$\begin{aligned} & \int \frac{\sin \frac{5 x}{2}}{\sin \frac{x}{2}} d x=\int \frac{\sin \left(\frac{x}{2}+2 xight)}{\sin \frac{x}{2}} d x \ & =\int \frac{\sin \frac{x}{2} \cos 2 x+\cos \frac{x}{2} \sin 2 x}{\sin \frac{x}{2}} d x \ & \int\left(\cos 2 x+\cos \frac{x}{2} \cdot 4 \cos \frac{x}{2} \cdot \cos xight) d x \ & =\int(\cos 2 x+2(1+\cos x) \cos x) d x \ & =\int\left(\cos 2 x+2 \cos x+2 \cos ^2 xight) d x \ & =\int(\cos 2 x+2 \cos x+1+\cos 2 x) d x\end{aligned}$ $\begin{aligned} & =\int(1+2 \cos x+2 \cos x) d x \ & =x+2 \sin x+\sin 2 x+C\end{aligned}$

$\int \frac{\sin \frac{5 x}{2}}{\sin \frac{x}{2}} d x=$ (where $C$ is a constant of integration.)

Solution