Question Detail
Question
Which of the following option best describes the reduction formula for \(V_n = \int \csc^n x\,dx\), where \(n \ge 2\)?
Correct Answer
Option A is the correct answer.
Detailed Explanation
The standard reduction formula for (\int \csc^n x\,dx) is given by:(\int \csc^n x\,dx = -\frac{\csc^{n-2}x \cot x}{n-1} + \frac{n-2}{n-1} \int \csc^{n-2}x\,dx)If we denote (V_n = \int \csc^n x\,dx), then the formula can be written as:(V_n = -\frac{1}{n-1}\csc^{n-2}x \cot x + \frac{n-2}{n-1}V_{n-2}).This matches option A.
Hint
Recall the reduction formula for powers of cosecant function.