Question

Integrate the rational function:

`(cos x)/((1-sinx)(2 - sin x))` [Hint: Put sin x = t]

Answer 1

Let `(cos x)/((1 - sin x)(2 - sin x))` dx

Put sin x = t 

cos x dx = dt

`therefore I = int dt/((1 - t)(2 - t))`      .... (1)

Let `1/((1 -  t)(2 -  t)) = A/(1 - t) + B/(2 - t)`

⇒ 1 = A (2 - t) = B (1 - t)        .... (2)

Putting t = 1 in equation (2),

1 = A (2 - 1)

⇒ A = 1

Putting t = 2 in equation (2),

1 = B (1 - 2)

⇒ B = -1

`therefore` from equation (1),

`int (cos x)/((1 - sin x)(2 - sin x)) = int 1/(1 -  t)  dt - int dt/(2 - t)`

`= -log abs(1 -  t) + log abs(2 -  t) + C`

`= log abs((2 -  t)/(1 - t)) + C = log abs ((2 - sin x)/(1 - sin x)) + C`