The value of `2intsinx." cosec "4x`dx is equal to

Options

A. `1/(4sqrt(2))"ln"|(1+sqrt(2)sinx)/(1-sqrt(2)sinx)|-1/4"ln"|(1+sinx)/(1-sinx)|+C`
B. `1/(2sqrt(2))"ln|(1+sqrt(2)sinx)/(1-sqrt(2)sinx)-1/2"ln"|(1+sinx)/(cosx)|+C`
C. `1/(2sqrt(2))"ln"|(1-sqrt(2)sinx)/(1+sqrt(2)sinx)|-1/4"ln"|(1+sinx)/(1-sinx)|+C`
D. `-1/(2sqrt(2))"ln"|(1-sqrt(2)sinx)/(1+sqrt(2)sinx)|+1/4"ln"|(1-sinx)/(1+sinx)|+C`

Explanation

To solve the integral \( 2 \int \sin x \cdot \csc(4x) \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ 2 \int \sin x \cdot \csc(4x) \, dx \]
Recall that \(\csc(4x) = \frac{1}{\sin(4x)}\), so we can rewrite the integral as:
...