`int ( n sin^(n-1))/(secx (sin x)^(n)) dx ` equals :

Options

A. `(cos^(n))/((1+sin x)^(n))`
B. `(sin ^(x)x)/((1+sin x)^(n))`
C. `(-cos^(x))/((1+sin x)^(n))+C`
D. none of these

Answer

To solve the integral

\[ \int \frac{n \sin^{n-1}(x)}{\sec(x) \sin^n(x)} \, dx, \]

we can simplify the expression and then find the solution step by step.

### Step 1: Rewrite the integral
We can rewrite \(\sec(x)\) in terms of \(\cos(x)\):

\[ \sec(x) = \frac{1}{\cos(x)}. \]

Thus, the integral becomes:

\[ \int \frac{n \sin^{n-1}(x)}{\frac{1}{\cos(x)} \sin^n(x)} \, dx = \int n \sin^{n-1}(x) \cos(x) \cdot \sin^{-n}(x) \, dx. \]

### Step 2: Simplify the integrand
Now, we can simplify the integrand:

\[ \int n \sin^{n-1}(x) \cos(x) \cdot \frac{1}{\sin^n(x)} \, dx = \int n \frac{\cos(x)}{\sin(x)} \, dx = n \int \cot(x) \, dx. \]

### Step 3: Integrate \(\cot(x)\)
The integral of \(\cot(x)\) is:

\[ \int \cot(x) \, dx = \ln|\sin(x)| + C. \]

Thus, we have:

\[ n \int \cot(x) \, dx = n \ln|\sin(x)| + C. \]

### Step 4: Final answer
Therefore, the final answer for the integral is:

\[ \int \frac{n \sin^{n-1}(x)}{\sec(x) \sin^n(x)} \, dx = n \ln|\sin(x)| + C. \]