`intsec^(5)xtanxdx=?`

Options

A. `(1)/(5)tan^(5)x+C`
B. `(1)/(5)sec^(5)x+C`
C. `5log|cosx|+C`
D. none of these

Explanation

To solve the integral \( \int \sec^5 x \tan x \, dx \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Integral**:
We start with the integral:
\[ I = \int \sec^5 x \tan x \, dx \]

2. **Substitution**:
We can use the substitution \( t = \sec x \). The derivative of \( \sec x \) is \( \sec x \tan x \), which gives us:
\[ dt = \sec x \tan x \, dx \quad \Rightarrow \quad dx = \frac{dt}{\sec x \tan x} \]

3. **Rewrite the Integral**:
Substitute \( \sec x = t \) into the integral:
\[ I = \int t^5 \cdot \sec x \tan x \, dx \]
Since \( \sec x \tan x \, dx = dt \), we can rewrite the integral as:
\[ I = \int t^5 \, dt \]

4. **Integrate**:
Now, we can integrate \( t^5 \):
\[ I = \frac{t^6}{6} + C \]

5. **Back Substitute**:
Replace \( t \) back with \( \sec x \):
\[ I = \frac{\sec^6 x}{6} + C \]

### Final Answer:
Thus, the integral \( \int \sec^5 x \tan x \, dx \) evaluates to:
\[ \int \sec^5 x \tan x \, dx = \frac{\sec^6 x}{6} + C \]