Write integrating factor differential equations
`(dy)/(dx)+(1)/(1+x^2)y= sin x`

Answer

To find the integrating factor for the differential equation

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

we will follow these steps:

### Step 1: Identify \( p(x) \) and \( q(x) \)

The given differential equation is in the standard form:

\[ \frac{dy}{dx} + p(x)y = q(x), \]

where \( p(x) = \frac{1}{1+x^2} \) and \( q(x) = \sin x \).

### Step 2: Calculate the Integrating Factor

The integrating factor \( IF \) is given by the formula:

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

Substituting \( p(x) \):

\[ IF = e^{\int \frac{1}{1+x^2} \, dx}. \]

### Step 3: Evaluate the Integral

The integral \( \int \frac{1}{1+x^2} \, dx \) is a standard integral that evaluates to:

\[ \int \frac{1}{1+x^2} \, dx = \tan^{-1}(x) + C, \]

where \( C \) is the constant of integration. Therefore, we can write:

\[ IF = e^{\tan^{-1}(x) + C} = e^{\tan^{-1}(x)} \cdot e^C. \]

Since \( e^C \) is a constant, we can denote it as \( k \). Thus, we have:

\[ IF = k \cdot e^{\tan^{-1}(x)}. \]

For the purpose of finding the integrating factor, we can ignore the constant \( k \) and write:

\[ IF = e^{\tan^{-1}(x)}. \]

### Step 4: Final Result

The integrating factor for the given differential equation is:

\[ IF = e^{\tan^{-1}(x)}. \]

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