The reduction formula of In=∫ cot n x d x is

Question

The reduction formula of In=∫ cot n x d x is (A) In=( cot n-1 x/n-1)-In-2, n ≥ 2 (B) In=-( cot n-1 x/n-1)-In-2, n ≥ 2 (C) In=-( cot n-1 x/n-1)+In-2, n ≥ 2 (D) In=( cot n-1 x/n-1)+In-2, n ≥ 2

Tardigrade Admin · Accepted Answer

In=∫ cot n x d x=∫ cot 2 x ⋅ cot n-2 x d x =∫( textcosec2 x-1) cot n-2 x d x ⇒ In=∫ textcosec2 x cot n-2 x d x-In-2 ⇒ In=-( cot n-1 x/n-1)-In-2, n ≥ 2

Q. The reduction formula of $I_{n} = \int cot^{n} x d x$ is

$I_{n} = \frac{c o t ^{n - 1} x}{n - 1} - I_{n - 2}, n \geq 2$

$I_{n} = - \frac{c o t ^{n - 1} x}{n - 1} - I_{n - 2}, n \geq 2$

$I_{n} = - \frac{c o t ^{n - 1} x}{n - 1} + I_{n - 2}, n \geq 2$

$I_{n} = \frac{c o t ^{n - 1} x}{n - 1} + I_{n - 2}, n \geq 2$

$I_{n} = \int cot^{n} x d x = \int cot^{2} x \cdot cot^{n - 2} x d x$
$= \int (cosec^{2} x - 1) cot^{n - 2} x d x$
$\Rightarrow I_{n} = \int cosec^{2} x cot^{n - 2} x d x - I_{n - 2}$
$\Rightarrow I_{n} = - \frac{c o t ^{n - 1} x}{n - 1} - I_{n - 2}, n \geq 2$

Q. The reduction formula of In​=∫cotnxdx is

In​=n−1cotn−1x​−In−2​,n≥2

In​=−n−1cotn−1x​−In−2​,n≥2

In​=−n−1cotn−1x​+In−2​,n≥2

In​=n−1cotn−1x​+In−2​,n≥2