An integrating factor of the differential equation
` sin x (dy)/(dx) + 2 y cos x =1 ` is

Options

A. `sin^2 x `
B. `(2)/(sin x) `
C. `log|sin x |`
D. `(1)/(sin^(2) x)`

Answer

To find the integrating factor of the differential equation

\[ \sin x \frac{dy}{dx} + 2y \cos x = 1, \]

we will follow these steps:

### Step 1: Rewrite the Differential Equation
First, we can rewrite the given differential equation in standard form. We divide the entire equation by \(\sin x\):

\[ \frac{dy}{dx} + 2y \frac{\cos x}{\sin x} = \frac{1}{\sin x}. \]

This simplifies to:

\[ \frac{dy}{dx} + 2y \cot x = \csc x. \]

### Step 2: Identify \(p(x)\) and \(q(x)\)
Now, we can identify \(p(x)\) and \(q(x)\) from the standard linear form \(\frac{dy}{dx} + p(x)y = q(x)\):

- Here, \(p(x) = 2 \cot x\)
- \(q(x) = \csc x\)

### Step 3: Find the Integrating Factor
The integrating factor (IF) is given by the formula:

\[ IF = e^{\int p(x) \, dx}. \]

Substituting \(p(x)\):

\[ IF = e^{\int 2 \cot x \, dx}. \]

### Step 4: Calculate the Integral
The integral of \(2 \cot x\) is:

\[ \int 2 \cot x \, dx = 2 \ln |\sin x| + C. \]

Thus, we have:

\[ IF = e^{2 \ln |\sin x|} = |\sin x|^2 = \sin^2 x. \]

### Step 5: Conclusion
Therefore, the integrating factor for the differential equation is:

\[ \sin^2 x. \]

---