`int tan^(2)" x dx "`

Answer

To solve the integral \( \int \tan^2 x \, dx \), we can use the identity that relates tangent to secant. The steps are as follows:

### Step 1: Use the identity for \(\tan^2 x\)
We know that:
\[ \tan^2 x = \sec^2 x - 1 \]
Thus, we can rewrite the integral:
\[ \int \tan^2 x \, dx = \int (\sec^2 x - 1) \, dx \]

### Step 2: Separate the integral
Now, we can separate the integral into two parts:
\[ \int \tan^2 x \, dx = \int \sec^2 x \, dx - \int 1 \, dx \]

### Step 3: Integrate each part
1. The integral of \(\sec^2 x\) is:
\[ \int \sec^2 x \, dx = \tan x \]
2. The integral of \(1\) is:
\[ \int 1 \, dx = x \]

### Step 4: Combine the results
Putting it all together, we have:
\[ \int \tan^2 x \, dx = \tan x - x + C \]
where \(C\) is the constant of integration.

### Final Answer
Thus, the final result is:
\[ \int \tan^2 x \, dx = \tan x - x + C \]

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