Question

$\int\frac{\sin \frac{5x}{2}}{\sin \frac{x}{2}} dx $ is equal to : (where $c$ is a constant of integration)

• A. $2x + sin \,x + 2 sin\,2x + c$
• B. $x + 2\,sinx + 2\,sin2x + c$
• C. $x + 2\, sin x + sin \,2x + c$
• D. $2x + sinx + sin2x + c$

Answer 1

The Correct Option is C

We need to evaluate the integral $\int \frac{\sin \frac{5x}{2}}{\sin \frac{x}{2}} \, dx$. To solve this, we can use the sine angle addition formula:

$ \sin(A + B) = \sin A \cos B + \cos A \sin B $.

By applying successive use of angle addition identities, we know that:

$ \sin \left(nx \right) = n\sin\left(\frac{x}{2} \right)\cos\left(\frac{(n-1)x}{2} \right) + (n-1)\cos\left(\frac{x}{2} \right)\sin\left(\frac{(n-1)x}{2} \right) $

In particular, we will use:

• $\sin \frac{5x}{2} = \sin(2x + \frac{x}{2})$
• By expanding each part, the above becomes:
• Total: $ \sin \left(2x\right)\cos\left(\frac{x}{2}\right) + (1)\sin \left(\frac{x}{2}\right) $

Thus,

$\frac{\sin \frac{5x}{2}}{\sin \frac{x}{2}} = 2 \cos x + 1$.

Hence, our integral becomes:

$\int \left( 2 \cos x + 1 \right) \, dx$ .

Now solve this integral:

1. Integrate $2 \cos x$ to get $2 \sin x$.
2. Integrate $1$ to get $x$.

Therefore, the integral is:

$x + 2 \sin x + c$ , where $c$ is a constant of integration.

This matches the correct answer:

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

Thus, the solution has been verified, and option '$x + 2\, sin x + sin \,2x + c$' is correct.