`int x sec^2x dx`

Answer

To solve the integral \( \int x \sec^2 x \, dx \), we will use the method of integration by parts.

### Step-by-Step Solution:

1. **Identify \( u \) and \( dv \)**:
We choose:
\[ u = x \quad \text{and} \quad dv = \sec^2 x \, dx \]

2. **Differentiate \( u \) and Integrate \( dv \)**:
Now we differentiate \( u \) and integrate \( dv \):
\[ du = dx \quad \text{and} \quad v = \int \sec^2 x \, dx = \tan x \]

3. **Apply the Integration by Parts Formula**:
The integration by parts formula is given by:
\[ \int u \, dv = uv - \int v \, du \]
Substituting our values:
\[ \int x \sec^2 x \, dx = x \tan x - \int \tan x \, dx \]

4. **Integrate \( \tan x \)**:
We know that:
\[ \int \tan x \, dx = -\ln |\cos x| + C \]
Therefore, substituting this back into our equation:
\[ \int x \sec^2 x \, dx = x \tan x - (-\ln |\cos x|) + C \]
Simplifying this gives:
\[ \int x \sec^2 x \, dx = x \tan x + \ln |\cos x| + C \]

### Final Answer:
\[ \int x \sec^2 x \, dx = x \tan x + \ln |\cos x| + C \]