The integrating factor of `cos^(2) x(dy)/(dx) +y = tan x` is

Options

A. `e^(sin x)`
B. `e^(cos x)`
C. `e^(tan x)`
D. `e^(cot x)`

Answer

To find the integrating factor for the differential equation given by

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

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, which is

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

To do this, we divide the entire equation by \(\cos^2 x\):

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

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

- \(P(x) = \frac{1}{\cos^2 x} = \sec^2 x\)
- \(Q(x) = \frac{\tan x}{\cos^2 x} = \sec^2 x \tan x\)

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

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

Substituting \(P(x)\):

\[ \mu(x) = e^{\int \sec^2 x \, dx}. \]

### Step 4: Calculate the integral
The integral of \(\sec^2 x\) is:

\[ \int \sec^2 x \, dx = \tan x + C, \]

where \(C\) is the constant of integration. Thus, we have:

\[ \mu(x) = e^{\tan x + C} = e^{\tan x} \cdot e^C. \]

Since \(e^C\) is just a constant, we can ignore it for the purpose of finding the integrating factor:

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

### Final Answer
Thus, the integrating factor is:

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

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