`int sec x (sec x + tanx ) dx = `

Options

A. `tanx - sec x + C`
B. `-tanx + secx + C`
C. `tanx + sec x + C`
D. `-tanx - secx + C`

Answer

To solve the integral \( \int \sec x (\sec x + \tan x) \, dx \), we can follow these steps:

### Step 1: Expand the integrand
First, we expand the expression inside the integral:
\[ \sec x (\sec x + \tan x) = \sec^2 x + \sec x \tan x \]
So, we rewrite the integral:
\[ \int \sec x (\sec x + \tan x) \, dx = \int (\sec^2 x + \sec x \tan x) \, dx \]

### Step 2: Split the integral
Now, we can split the integral into two separate integrals:
\[ \int (\sec^2 x + \sec x \tan x) \, dx = \int \sec^2 x \, dx + \int \sec x \tan x \, dx \]

### Step 3: Integrate each part
We know the standard integrals:
- The integral of \( \sec^2 x \) is \( \tan x \).
- The integral of \( \sec x \tan x \) is \( \sec x \).

So we can write:
\[ \int \sec^2 x \, dx = \tan x \]
\[ \int \sec x \tan x \, dx = \sec x \]

### Step 4: Combine the results
Now, we combine the results of the two integrals:
\[ \int \sec^2 x \, dx + \int \sec x \tan x \, dx = \tan x + \sec x + C \]
where \( C \) is the constant of integration.

### Final Answer
Thus, the final result of the integral is:
\[ \int \sec x (\sec x + \tan x) \, dx = \tan x + \sec x + C \]

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