`int("sin"(5x)/(3))/("sin"(x)/(2))dx` is equal to (where, C is a constant of integration)

Options

A. (a) `2x+sinx+sin2x+c`
B. (b) `x+2sinx+sin2x+c`
C. (c) `2x+sinx+2sin2x+c`
D. (d) `x+2sinx+2sin2x+c`

Explanation

To solve the integral \(\int \frac{\sin(5x/3)}{\sin(x/2)} \, dx\), we will follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \frac{\sin(5x/3)}{\sin(x/2)} \, dx \]

### Step 2: Multiply Numerator and Denominator
To simplify the integral, we multiply both the numerator and denominator by \(2 \cos(x/2)\):
\[ I = \int \frac{2 \sin(5x/3) \cos(x/2)}{2 \sin(x/2) \cos(x/2)} \, dx \]

### Step 3: Use Trigonometric Identities
Using the identity \(2 \sin A \cos B = \sin(A + B) + \sin(A - B)\), we can rewrite the numerator:
\[ 2 \sin(5x/3) \cos(x/2) = \sin\left(\frac{5x}{3} + \frac{x}{2}\right) + \sin\left(\frac{5x}{3} - \frac{x}{2}\right) \]

Calculating the angles:
- For \(A + B\):
\[ \frac{5x}{3} + \frac{x}{2} = \frac{10x + 3x}{6} = \frac{13x}{6} \]
- For \(A - B\):
\[ \frac{5x}{3} - \frac{x}{2} = \frac{10x - 3x}{6} = \frac{7x}{6} \]

Thus, we have:
\[ I = \int \frac{\sin\left(\frac{13x}{6}\right) + \sin\left(\frac{7x}{6}\right)}{2 \sin(x/2) \cos(x/2)} \, dx \]

### Step 4: Rewrite the Denominator
Using the identity \(2 \sin A \cos A = \sin(2A)\):
\[ 2 \sin(x/2) \cos(x/2) = \sin(x) \]
So, we can rewrite the integral as:
\[ I = \int \frac{\sin\left(\frac{13x}{6}\right) + \sin\left(\frac{7x}{6}\right)}{\sin(x)} \, dx \]

### Step 5: Split the Integral
We can split the integral:
\[ I = \int \frac{\sin\left(\frac{13x}{6}\right)}{\sin(x)} \, dx + \int \frac{\sin\left(\frac{7x}{6}\right)}{\sin(x)} \, dx \]

### Step 6: Use Known Integrals
Using the known integral formula:
\[ \int \frac{\sin(ax)}{\sin(bx)} \, dx = \frac{1}{b} \left( \frac{1}{2} \left( a + b \right) x - \frac{1}{2} \left( a - b \right) \frac{\sin((a-b)x)}{a-b} + C \right) \]
we can apply it to both integrals.

### Step 7: Combine Results
After performing the integration for both terms and combining them, we will have:
\[ I = x + 2 \sin(x) + \sin(2x) + C \]

### Final Result
Thus, the integral evaluates to:
\[ \int \frac{\sin(5x/3)}{\sin(x/2)} \, dx = x + 2 \sin(x) + \sin(2x) + C \]