The value of the integral ` int_(0)^(a//2) sin 2n x cot x dx,` where n is a positive integer, is

Options

A. `(pi)/(2)`
B. `-pi`
C. `pi`
D. none of these

Answer

A

Explanation

We have,
`cos x +cos 3x +cos 5x+ ...+ cos (2n-1) x=(cos[ x+(n-1)x] sin n x)/(sin x)`
`rArr 2 [ cos x+ cos 3x +cos 5x+...+ cos (2n-1)x]= (sin 2nx)/(sinx)`
`rArr sin 2x cos x = 2 cos x { cos x+cos 3x+cos 5x+....+ cos (2n-1)x}`
`rArr sin 2 x cot x= 1 +2 {cos 2x+cos4x +cox 6x +...+ cos (2n-2)x} +cos 2nx`
`rArr underset(0)overset(pi//2)int sin 2nx cot x dx =(pi)/(2)+2 underset(0)overset(pi//2)int {c os 2x+c os 4x+....+ cos (2n-2)x} dx+underset(0)overset(pi//2)int cos 2nx dx`
`rArr underset(0)overset(pi//2)int sin 2nx cot x dx =(pi)/(2)+2 xx0+0=(pi)/(2)`