Given `(sin2kx)/(sin x)=2[cos x+cos 3x+…+ cos(2k-1)x]`, where k is a positive integer, show that,
`int_(0)^((pi)/(2))sin2 kx cotx dx=(pi)/(2)`.

int_(0)^( pi/2)sin^(2)x cos x dx

int_(0)^(pi/2)(sin^(2)x*cos x)dx=

int_(0)^(pi) dx/(3+2sin x + cos x) =

int_(0)^( pi/2)(sin x)/(1+cos x)dx

Prove that for any positive integer K, (sin2kx)/(sinx)=2[cos x + cos 3x+....+ cos (2k-1)x] Hence , proved that int_(0)^(pi//2) sin 2kx. Cot x dx = (pi//2)

Prove that for any positive integer k,(sin2kx)/(sin x)=2[cos x+cos3x+...+cos(2k-1)x] Hence,prove that int_(0)^((pi)/(2))sin2xk cot xdx=(pi)/(2)

int_(-pi)^(pi)sin^(2)x.cos^(2)x dx=

(sin 2 kx)/(sin x)=2[cos x + cos 3 x + …+cos (2k -1)x] , then value of I=int_(0)^(pi//2)(sin 2 k x)/(sin x)cos x dx is :

(sin 2 kx)/(sin x)=2[cos x + cos 3 x + …+cos (2k -1)x] , then value of I=int_(0)^(pi//2)(sin 2 k x)/(sin x)cos x dx is :

int_(0)^(pi//2) (sin x)/(1 +cos^2) dx .