Find the integrating factor of the differential equation :
`cos x(dy)/(dx)+y=2x+x^(2)`

Answer

To find the integrating factor of the differential equation given by

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

we will follow these steps:

### Step 1: Rewrite the equation in standard form
We start by rewriting the equation in the standard form of a linear first-order differential equation:

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

### Step 2: Identify \( p(x) \) and \( q(x) \)
From the rewritten equation, we can identify:

- \( p(x) = \frac{1}{\cos x} = \sec x \)
- \( q(x) = \frac{2x + x^2}{\cos x} \)

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

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

Substituting \( p(x) \):

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

### Step 4: Calculate the integral of \( \sec x \)
The integral of \( \sec x \) is known to be:

\[ \int \sec x \, dx = \ln |\sec x + \tan x| + C. \]

Thus, we have:

\[ \mu(x) = e^{\ln |\sec x + \tan x|} = |\sec x + \tan x|. \]

Since the integrating factor is typically taken as a positive function, we can write:

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

### Final Answer
The integrating factor of the differential equation is:

\[ \sec x + \tan x. \]

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