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Question
Integrate the rational function:
`(cos x)/((1-sinx)(2 - sin x))` [Hint: Put sin x = t]
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Solution
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`
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