Evaluate `int tan^(3)x dx`

Explanation

To evaluate the integral \( \int \tan^3 x \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start by rewriting \( \tan^3 x \) in terms of \( \tan^2 x \):
\[ \int \tan^3 x \, dx = \int \tan^2 x \tan x \, dx \]
Using the identity \( \tan^2 x = \sec^2 x - 1 \), we can express the integral as:
\[ \int \tan^3 x \, dx = \int (\sec^2 x - 1) \tan x \, dx \]

### Step 2: Split the Integral
Now, we can split the integral into two parts:
\[ \int \tan^3 x \, dx = \int \sec^2 x \tan x \, dx - \int \tan x \, dx \]

### Step 3: Evaluate the First Integral
The first integral \( \int \sec^2 x \tan x \, dx \) can be evaluated using the substitution \( u = \tan x \). Then, \( du = \sec^2 x \, dx \), which gives:
\[ \int \sec^2 x \tan x \, dx = \int u \, du = \frac{u^2}{2} + C = \frac{\tan^2 x}{2} + C \]

### Step 4: Evaluate the Second Integral
The second integral \( \int \tan x \, dx \) is a standard integral:
\[ \int \tan x \, dx = -\log |\cos x| + C = \log |\sec x| + C \]

### Step 5: Combine the Results
Now, substituting back into our expression, we have:
\[ \int \tan^3 x \, dx = \frac{\tan^2 x}{2} - \log |\sec x| + C \]

### Final Result
Thus, the final result for the integral \( \int \tan^3 x \, dx \) is:
\[ \int \tan^3 x \, dx = \frac{\tan^2 x}{2} - \log |\sec x| + C \]