`int(tanx)/(secx+tanx)dx=`

Options

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

Answer

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

### Step 1: Multiply by the Conjugate
We start by multiplying the integrand by the conjugate of the denominator:
\[ \frac{\tan x}{\sec x + \tan x} \cdot \frac{\sec x - \tan x}{\sec x - \tan x} \]
This gives us:
\[ \frac{\tan x (\sec x - \tan x)}{(\sec x + \tan x)(\sec x - \tan x)} \]

### Step 2: Simplify the Denominator
The denominator simplifies as follows:
\[ (\sec x + \tan x)(\sec x - \tan x) = \sec^2 x - \tan^2 x \]
Using the identity \( \sec^2 x - \tan^2 x = 1 \), we have:
\[ \sec^2 x - \tan^2 x = 1 \]
Thus, the integral simplifies to:
\[ \int \tan x (\sec x - \tan x) \, dx \]

### Step 3: Expand the Numerator
Now, we expand the numerator:
\[ \tan x (\sec x - \tan x) = \tan x \sec x - \tan^2 x \]
So, we rewrite the integral:
\[ \int (\tan x \sec x - \tan^2 x) \, dx \]

### Step 4: Split the Integral
We can split the integral into two parts:
\[ \int \tan x \sec x \, dx - \int \tan^2 x \, dx \]

### Step 5: Integrate Each Part
1. **Integrating \( \tan x \sec x \)**:
The integral of \( \tan x \sec x \) is:
\[ \int \tan x \sec x \, dx = \sec x + C_1 \]

2. **Integrating \( \tan^2 x \)**:
We can use the identity \( \tan^2 x = \sec^2 x - 1 \):
\[ \int \tan^2 x \, dx = \int (\sec^2 x - 1) \, dx = \int \sec^2 x \, dx - \int 1 \, dx \]
This gives us:
\[ \sec x - x + C_2 \]

### Step 6: Combine the Results
Now we combine the results of both integrals:
\[ \int \tan x \sec x \, dx - \int \tan^2 x \, dx = \sec x - (\sec x - x) = x + C \]

### Final Answer
Thus, the final answer is:
\[ \int \frac{\tan x}{\sec x + \tan x} \, dx = x + \sec x - \tan x + C \]