`int((1)/(cos^(2)x)-(1)/(sin^(2)x))dx`

Options

A. `tanx+cotx+C`
B. `cotx-tanx+C`
C. `-cotx-tanx+C`
D. `cotx+tanx+C`

Answer

To solve the integral \(\int \left( \frac{1}{\cos^2 x} - \frac{1}{\sin^2 x} \right) dx\), we can follow these steps:

### Step 1: Rewrite the integrand
We can rewrite the integrand using the identities for secant and cosecant:
\[ \frac{1}{\cos^2 x} = \sec^2 x \quad \text{and} \quad \frac{1}{\sin^2 x} = \csc^2 x \]
Thus, the integral becomes:
\[ \int \left( \sec^2 x - \csc^2 x \right) dx \]

### Step 2: Separate the integral
We can separate the integral into two parts:
\[ \int \sec^2 x \, dx - \int \csc^2 x \, dx \]

### Step 3: Integrate each term
Now we can integrate each term separately:
1. The integral of \(\sec^2 x\) is:
\[ \int \sec^2 x \, dx = \tan x \]
2. The integral of \(\csc^2 x\) is:
\[ \int \csc^2 x \, dx = -\cot x \]

### Step 4: Combine the results
Combining the results from the two integrals, we have:
\[ \tan x - (-\cot x) = \tan x + \cot x \]

### Step 5: Add the constant of integration
Finally, we add the constant of integration \(C\):
\[ \int \left( \frac{1}{\cos^2 x} - \frac{1}{\sin^2 x} \right) dx = \tan x + \cot x + C \]

### Final Answer
Thus, the final answer is:
\[ \tan x + \cot x + C \]

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