`int(dx)/((1+cos^(2)x))`

Explanation

To solve the integral \( \int \frac{dx}{1 + \cos^2 x} \), we will follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \frac{dx}{1 + \cos^2 x} \]

### Step 2: Divide by \( \cos^2 x \)
We can simplify the integrand by dividing both the numerator and the denominator by \( \cos^2 x \):
\[ I = \int \frac{1}{\cos^2 x} \cdot \frac{dx}{\frac{1}{\cos^2 x} + 1} \]
This gives us:
\[ I = \int \frac{\sec^2 x \, dx}{\sec^2 x + 1} \]

### Step 3: Simplify the Denominator
Recall that \( \sec^2 x = 1 + \tan^2 x \). Thus, we can rewrite the denominator:
\[ \sec^2 x + 1 = 1 + \tan^2 x + 1 = \tan^2 x + 2 \]
So now our integral becomes:
\[ I = \int \frac{\sec^2 x \, dx}{\tan^2 x + 2} \]

### Step 4: Use Substitution
Let \( t = \tan x \). Then, \( dt = \sec^2 x \, dx \). The integral now transforms to:
\[ I = \int \frac{dt}{t^2 + 2} \]

### Step 5: Recognize the Standard Integral
The integral \( \int \frac{dt}{t^2 + a^2} \) has a standard solution:
\[ \int \frac{dt}{t^2 + a^2} = \frac{1}{a} \tan^{-1} \left( \frac{t}{a} \right) + C \]
In our case, \( a^2 = 2 \) implies \( a = \sqrt{2} \):
\[ I = \frac{1}{\sqrt{2}} \tan^{-1} \left( \frac{t}{\sqrt{2}} \right) + C \]

### Step 6: Substitute Back
Now we substitute back \( t = \tan x \):
\[ I = \frac{1}{\sqrt{2}} \tan^{-1} \left( \frac{\tan x}{\sqrt{2}} \right) + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int \frac{dx}{1 + \cos^2 x} = \frac{1}{\sqrt{2}} \tan^{-1} \left( \frac{\tan x}{\sqrt{2}} \right) + C \]