`int(1+tan^(2)x)/(1+cot^(2)x)dx-`

Options

A. `sinx-cos x+c`
B. `secx-cos x+c`
C. `tanx-cotx+c`
D. `tanx-x+c`

Answer

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

### Step 1: Simplify the integrand
We know from trigonometric identities that:
- \( 1 + \tan^2 x = \sec^2 x \)
- \( 1 + \cot^2 x = \csc^2 x \)

Thus, we can rewrite the integral as:
\[ \int \frac{\sec^2 x}{\csc^2 x} \, dx \]

### Step 2: Rewrite in terms of sine and cosine
Using the definitions of secant and cosecant:
\[ \sec^2 x = \frac{1}{\cos^2 x} \quad \text{and} \quad \csc^2 x = \frac{1}{\sin^2 x} \]

So, we have:
\[ \frac{\sec^2 x}{\csc^2 x} = \frac{\frac{1}{\cos^2 x}}{\frac{1}{\sin^2 x}} = \frac{\sin^2 x}{\cos^2 x} \]

This simplifies to:
\[ \tan^2 x \]

### Step 3: Set up the new integral
Now, our integral becomes:
\[ \int \tan^2 x \, dx \]

### Step 4: Use the identity for tangent squared
Recall the identity:
\[ \tan^2 x = \sec^2 x - 1 \]

Thus, we can rewrite the integral as:
\[ \int (\sec^2 x - 1) \, dx \]

### Step 5: Separate the integral
Now we can separate the integral:
\[ \int \sec^2 x \, dx - \int 1 \, dx \]

### Step 6: Integrate each part
The integral of \( \sec^2 x \) is:
\[ \tan x \]
And the integral of \( 1 \) is:
\[ x \]

Thus, we have:
\[ \tan x - x + C \]

### Final Answer
The final result of the integral is:
\[ \int \frac{1 + \tan^2 x}{1 + \cot^2 x} \, dx = \tan x - x + C \]

---