` int (3-4 sin x)/(cos^(2) x) dx=?`

Options

A. `4 tan x +4 sec x+c`
B. ` 3 tan x + 4 sec x+ c`
C. ` 4 tan x +3 sec x+c`
D. ` 3 tan x - 4 sec x+ c`

Answer

To solve the integral \( \int \frac{3 - 4 \sin x}{\cos^2 x} \, dx \), we can follow these steps:

### Step 1: Separate the Integral
We can separate the integral into two parts:
\[ \int \frac{3 - 4 \sin x}{\cos^2 x} \, dx = \int \frac{3}{\cos^2 x} \, dx - \int \frac{4 \sin x}{\cos^2 x} \, dx \]

### Step 2: Rewrite the Integrals
We know that \( \frac{1}{\cos^2 x} = \sec^2 x \) and \( \frac{\sin x}{\cos^2 x} = \tan x \sec x \). Therefore, we can rewrite the integrals as:
\[ \int \frac{3}{\cos^2 x} \, dx = 3 \int \sec^2 x \, dx \]
\[ \int \frac{4 \sin x}{\cos^2 x} \, dx = 4 \int \tan x \sec x \, dx \]

### Step 3: Integrate Each Part
Now we can integrate each part:
1. The integral of \( \sec^2 x \) is \( \tan x \):
\[ 3 \int \sec^2 x \, dx = 3 \tan x \]
2. The integral of \( \tan x \sec x \) is \( \sec x \):
\[ 4 \int \tan x \sec x \, dx = 4 \sec x \]

### Step 4: Combine the Results
Combining the results from the two integrals, we get:
\[ \int \frac{3 - 4 \sin x}{\cos^2 x} \, dx = 3 \tan x - 4 \sec x + C \]
where \( C \) is the constant of integration.

### Final Answer
Thus, the final result is:
\[ \int \frac{3 - 4 \sin x}{\cos^2 x} \, dx = 3 \tan x - 4 \sec x + C \]