`int(sec^(8)x)/("cosec x")dx=`

Options

A. `(sec^(8)x)/(8)+c`
B. `(sec^(7)x)/(7)+c`
C. `(sec^(6)x)/(6)+c`
D. `(sec^(9)x)/(9)+c`

Explanation

To solve the integral \( \int \frac{\sec^8 x}{\csc x} \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start by rewriting the integral in terms of sine and cosine:
\[ \int \frac{\sec^8 x}{\csc x} \, dx = \int \frac{\sec^8 x}{\frac{1}{\sin x}} \, dx = \int \sec^8 x \cdot \sin x \, dx \]

### Step 2: Substitute for \(\sec x\)
Recall that \( \sec x = \frac{1}{\cos x} \). Thus, we can rewrite the integral as:
\[ \int \left(\frac{1}{\cos x}\right)^8 \sin x \, dx = \int \frac{\sin x}{\cos^8 x} \, dx \]

### Step 3: Use Substitution
Let \( t = \cos x \). Then, the derivative \( dt = -\sin x \, dx \) implies that \( dx = \frac{dt}{-\sin x} \). Substituting this into the integral gives:
\[ \int \frac{\sin x}{t^8} \cdot \frac{dt}{-\sin x} = -\int \frac{1}{t^8} \, dt \]

### Step 4: Integrate
Now we can integrate:
\[ -\int t^{-8} \, dt = -\left(\frac{t^{-7}}{-7}\right) = \frac{t^{-7}}{7} + C = \frac{1}{7t^7} + C \]

### Step 5: Substitute Back
Now we substitute back \( t = \cos x \):
\[ \frac{1}{7 \cos^7 x} + C \]

### Step 6: Convert to Secant
Since \( \sec x = \frac{1}{\cos x} \), we can rewrite the result as:
\[ \frac{\sec^7 x}{7} + C \]

### Final Answer
Thus, the final answer is:
\[ \int \frac{\sec^8 x}{\csc x} \, dx = \frac{\sec^7 x}{7} + C \]