Question Detail
Question
Which of the following option best describes the reduction formula for $V_n = \int \text{cosec}^n x dx$, $n \geq 2$.
Correct Answer
Option A is the correct answer.
Detailed Explanation
The standard reduction formula for the integral of $\text{cosec}^n x$ is derived using integration by parts.It is given by: $V_n = \int \text{cosec}^n x dx = -\frac{\text{cosec}^{n-2}x \cot x}{n-1} + \frac{n-2}{n-1} \int \text{cosec}^{n-2}x dx$.This can be written as $V_n = -\frac{1}{n-1} \text{cosec}^{n-2}x \cot x + \frac{n-2}{n-1} V_{n-2}$.Comparing this with the given options, option A exactly matches the standard reduction formula.
Hint
Recall the reduction formula for $\int \text{cosec}^n x dx$ which is derived using integration by parts, often by splitting $\text{cosec}^n x$ into $\text{cosec}^{n-2}x \cdot \text{cosec}^2 x$.