Integrate the following :
`int tan^(2)theta d theta.`

Explanation

To solve the integral \( \int \tan^2 \theta \, d\theta \), we can use a trigonometric identity and integration techniques. Here’s a step-by-step solution:

### Step 1: Use the identity for \( \tan^2 \theta \)
We know that:
\[ \tan^2 \theta = \sec^2 \theta - 1 \]
This allows us to rewrite the integral:
\[ \int \tan^2 \theta \, d\theta = \int (\sec^2 \theta - 1) \, d\theta \]

### Step 2: Split the integral
Now we can split the integral into two parts:
\[ \int \tan^2 \theta \, d\theta = \int \sec^2 \theta \, d\theta - \int 1 \, d\theta \]

### Step 3: Integrate each part
The integral of \( \sec^2 \theta \) is a standard integral:
\[ \int \sec^2 \theta \, d\theta = \tan \theta \]
And the integral of \( 1 \) is simply:
\[ \int 1 \, d\theta = \theta \]

### Step 4: Combine the results
Now we can combine the results from the two integrals:
\[ \int \tan^2 \theta \, d\theta = \tan \theta - \theta + C \]
where \( C \) is the constant of integration.

### Final Answer
Thus, the integral of \( \tan^2 \theta \) is:
\[ \int \tan^2 \theta \, d\theta = \tan \theta - \theta + C \]

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