Evaluate : `int (sin x)/((1+cos x)(2+3 cos x)) dx`

Options

A. `log |(2+cos x)/(2+ 3 cos x)| +c`
B. `log |(1+cos x)/(4+ 3 cos x)| +c`
C. `log |(1+cos x)/(2+ 3 cos x)| +c`
D. `log |(1+2cos x)/(2+ 3 cos x)| +c`

Answer

C

Explanation

`"Let I "= int (sinx)/((1+cos x) (2+ 3 cos x)) dx`
`=int (-dt)/((1+t)(2+3t))`
`"Let cos " x=t`
` -sin x =(dt)/(dx)`
` sin x dx =- dt`
`"Let " (-1)/((1+t)(2+3t))=(A)/(1+t)+(B)/(2+3t)`
`(A(2+3t)+B(1+t))/((1+t)(2+3t))`
`rArr A (2+3t)+B (1+t) =-1`
`"Put " t=-1`
`A(2-3) +0=1`
`rArr " "A=1`
` "Put " t=(2)/(3)`
`0+B(1-(2)/(3)) =-1`
` rArr " "B=-3`
`:. I= int((1)/(1+t)-(3)/(2+3t)) dt`
`=log |1+t|-log |2+ 3t|+c`
`= log |(1+t)/(2+3t)| +c`
`= log |(1+cos x)/(2+ 3 cos x)| +c`