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

Answer

To solve the integral \( \int x \sec^2 x \, dx \), we will use the method of integration by parts. The formula for integration by parts is given by:

\[ \int u \, dv = uv - \int v \, du \]

### Step-by-Step Solution:

1. **Choose \( u \) and \( dv \)**:
- Let \( u = x \) (which means \( du = dx \))
- Let \( dv = \sec^2 x \, dx \) (which means \( v = \tan x \))

2. **Apply the integration by parts formula**:
\[ \int x \sec^2 x \, dx = x \tan x - \int \tan x \, dx \]

3. **Integrate \( \tan x \)**:
- We know that \( \int \tan x \, dx = -\log |\cos x| + C \) (or equivalently \( \log |\sec x| + C \))

4. **Substitute back into the equation**:
\[ \int x \sec^2 x \, dx = x \tan x - (-\log |\cos x|) + C \]
This simplifies to:
\[ \int x \sec^2 x \, dx = x \tan x + \log |\cos x| + C \]

5. **Final answer**:
\[ \int x \sec^2 x \, dx = x \tan x + \log |\cos x| + C \]