Integrate the rational functions `(cosx)/((1-sinx)(2-sinx)` [Hint: Put `s in" "x" "=" "t` ]

Explanation

`" Let " I= int(cos x)/((1-sin x)(2-sinx))dx`
`" Let sin x=t " rArr cos x =(dt)/(dx) rArr cos dx=dt`
`:. int(1)/((1-t)(2-t))dx`
`" Let " (1)/((1-t)(2-t))=(A)/(1-t)+(B)/(2-t)" ".......(1)`
`rArr (1)/((1-t)(2-t)) =(A(2-t)+B(1-t))/((1-t)(2-t))`
`rArr 1=A(2-t)+B(1-t)`
`t=1 " then " 1=A(2-1) +0 rArr A=1`
`t=2 " then " 1=0 + B(1-2) rArr B=-1`
`:. I= int((1)/(1-t)-(1)/(2-t)) dt" " "[From eq. (1)]"`
` = int(1)/((1-t))dt - int(1)/((2-t))dt`
` =(log |1-t|)/((-1)) -(log |2-t|)/((-1))+c`
` = log |(2-t)/(1-t)| +C= log |(2-sin x)/(1-sin x)|+c`