1. We need to evaluate $I=\int \frac{\sin\left(\frac{5x}{2}\right)}{\sin\left(\frac{x}{2}\right)}\,dx.$

2. Use the standard identity $\frac{\sin\left(\left(n+\tfrac12\right)x\right)}{\sin\left(\frac{x}{2}\right)}=1+2\sum_{k=1}^{n}\cos(kx).$

Here, $\frac{5x}{2}=\left(2+\frac12\right)x,$ so $n=2$. Therefore, $\frac{\sin\left(\frac{5x}{2}\right)}{\sin\left(\frac{x}{2}\right)}=1+2\cos x+2\cos 2x.$

3. Hence the integral becomes $I=\int (1+2\cos x+2\cos 2x)\,dx.$

4. Integrate term-by-term: $\int 1\,dx=x,$ $\int 2\cos x\,dx=2\sin x,$ $\int 2\cos 2x\,dx=2\cdot \frac{\sin 2x}{2}=\sin 2x.$

So, $I=x+2\sin x+\sin 2x+c.$

5. Compare with the options:

• A: $2x+\sin x+2\sin 2x+c$
• B: $x+2\sin x+\sin 2x+c$
• C: $x+2\sin x+2\sin 2x+c$
• D: $2x+\sin x+\sin 2x+c$

Thus the correct option is $\boxed{\text{B}}.$

6. Verification with stored answer: Stored correct answer = B, which matches our derived answer.