Question

Integral (1+cosecx)*1/2

Answer 1

Answer:

2 sin^-1( sqrt( sin x)) + C

Step-by-step explanation:

∫ sqrt( 1+ csc x) dx = ∫ sqrt( 1+ 1/sin x) dx

= ∫ sqrt( sin x+ 1) / sqrt(sin x) dx

Let t= sqrt(sin x)

t^2 = sin x

2t dt = cos x dx

dx = 2 t dt / cos x

dx = 2 t dt / sqrt( 1-sin^2 x)

dx = 2t dt / sqrt( 1-t^4)

∫ sqrt( sin x+ 1) / sqrt(sin x) dx =2 ∫ sqrt( t^2+1) t dt / (sqrt(1-t^4) sqrt(t^2) )

=2 ∫ sqrt( t^2+1) dt / sqrt(1-t^4)

=2 ∫ sqrt( t^2+1) dt / sqrt [(1-t^2)(1+t^2)]

= 2 ∫ sqrt( t^2+1) dt / sqrt (1-t^2) sqrt (1+t^2)

= 2 ∫ dt / sqrt(1-t^2)

= 2 sin^-1( t)

replace t by sqrt(sin x)

= 2 sin^-1( sqrt( sin x)) + C