Question

The reduction formula of $I_n=\int \cot ^n x d x$ is

• A. $I_n=\frac{\cot ^{n-1} x}{n-1}-I_{n-2}, n \geq 2$
• B. $I_n=-\frac{\cot ^{n-1} x}{n-1}-I_{n-2}, n \geq 2$
• C. $I_n=-\frac{\cot ^{n-1} x}{n-1}+I_{n-2}, n \geq 2$
• D. $I_n=\frac{\cot ^{n-1} x}{n-1}+I_{n-2}, n \geq 2$

Answer 1

Answer: (B)

$I_n=\int \cot ^n x d x=\int \cot ^2 x \cdot \cot ^{n-2} x d x$
$=\int\left(\text{cosec}^2 x-1\right) \cot ^{n-2} x d x$
$\Rightarrow I_n=\int \text{cosec}^2 x \cot ^{n-2} x d x-I_{n-2}$
$\Rightarrow I_n=-\frac{\cot ^{n-1} x}{n-1}-I_{n-2}, n \geq 2$