Question

Answer of cosec 2x dx

Answer 1

Heya
We'll write the function cosec 2x = 1/sin 2x

We'll use the identity of double angle:

sin 2x = 2sin x*cos x

Int cosec 2x dx  =Int dx/sin 2x

Int dx/sin 2x = Int dx/2sinx*cosx

We'll use Pythagoras identity:

(sin x)^2 + (cos x)^2 = 1

Int dx/2sinx*cosx = Int [(sin x)^2 + (cos x)^2]dx/2sinx*cosx

Int dx/2sinx*cosx =(1/2)*[Int sin xdx/cos x + Int cos xdx/sin x]

(1/2)*[Int sin xdx/cos x + Int cos xdx/sin x] = (1/2)[-ln(cosx) + ln(sin x)]

Int cosec 2x dx = -ln(sqrt (cos x)) + ln (sqrt(sin x)) + C
Hope it helps