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

Options

A. cos x
B. tan x
C. sec x
D. sin x

Explanation

To find the integrating factor of the differential equation given by

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

we will follow these steps:

### Step 1: Rewrite the equation in standard form
First, we need to rewrite the equation in the standard linear form:

\[ \frac{dy}{dx} + P(x)y = Q(x). \]

To do this, we divide the entire equation by \(\cos x\):

\[ \frac{dy}{dx} + y \tan x = \sec x. \]

Here, we identify \(P(x) = \tan x\) and \(Q(x) = \sec x\).

### Step 2: Identify the integrating factor
The integrating factor \(\mu(x)\) is given by the formula:

\[ \mu(x) = e^{\int P(x) \, dx}. \]

Substituting \(P(x) = \tan x\):

\[ \mu(x) = e^{\int \tan x \, dx}. \]

### Step 3: Calculate the integral
The integral of \(\tan x\) is:

\[ \int \tan x \, dx = -\ln |\cos x| + C, \]

where \(C\) is a constant of integration. Therefore, we can express the integrating factor as:

\[ \mu(x) = e^{-\ln |\cos x|} = \frac{1}{|\cos x|}. \]

Since \(\cos x\) is positive in the intervals we typically consider, we can simplify this to:

\[ \mu(x) = \sec x. \]

### Step 4: Conclusion
Thus, the integrating factor of the given differential equation is:

\[ \mu(x) = \sec x. \]