Question

Integrate 1/sinx(5-4cosx)

Answer 1

Integrate:    (5-4 cos x) / sin x  dx

I = integral  [ 5 * (1 - cos x) / sin x ] dx    +  integral [ cos x/ sinx ]  dx
  = 5* Integral  2 sin² x/2  / (2 sin x/2  cos x/2) dx  +   integral  cotx dx
  =  5 * 2* Ln Sec  x/2  + Ln sin x  + K

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If the integral is like:

f(x) = 1/ [ sin x * (5- 4 cos x) ]

Let cos x = t,   dx =  -dt /√(1-t²) 

f(t) = - dt / [ (1-t^2) * (5 -t) ]
      = 1/24 *  5 dt/(1-t^2)   -   1/24 * t dt /(1- t^2)   + 1/24 * dt /(5-t)

Integral = 5/48  * Ln [(1+t)/(1-t) ]   + 1/48 * Ln (1-t^2)   -  1/24 * Ln | 5-t |   + K
       = 5/24 * Cot x/2   + 1/24 * Ln | sin x |  - 1/24  * Ln (5 - cosx )  + K