Find the integrating factor of the differential equation :
`cos x .(dy)/(dx)+y=sinx, 0 le x lt pi/2`

Answer

To find the integrating factor of the given differential equation \( \cos x \frac{dy}{dx} + y = \sin x \), we will follow these steps:

### Step 1: Rewrite the Differential Equation
We start by rewriting the differential equation in standard form. We can divide the entire equation by \( \cos x \):

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

This simplifies to:

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

### Step 2: Identify \( P(x) \) and \( Q(x) \)
From the standard form \( \frac{dy}{dx} + P(x) y = Q(x) \), we identify:

- \( P(x) = \tan x \)
- \( Q(x) = \tan x \)

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

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

Substituting \( P(x) \):

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

### Step 4: Compute the Integral
The integral of \( \tan x \) is:

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

Thus, we have:

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

### Step 5: Final Result
The integrating factor is:

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

### Summary
The integrating factor of the differential equation \( \cos x \frac{dy}{dx} + y = \sin x \) is \( \sec x \).

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