The value of `int(2x+2)/((x-2)^(2)(x-3))dx` is

Options

A. `11 log (x-3)/(x-2)-(8)/(x-2)`
B. `11 log (x+3)/(x+2)-(8)/(x-2)+c`
C. `11log (x-3)/(x-2)+(8)/(x-2)+c`
D. `11log (x+30)/(x+2)+(8)/(x-2)+c`

Answer

C

Explanation

Let `l =int(3x+2)/((x-2)^(2)(x-3))dx " "...(i)`
Again, let `(3x+2)/((x-2)(x-3))=(A)/((x-2))`
`+(B)/((x-2)^(2))+(C)/((x-3))" "...(ii)`
`implies3x+2=A(x-2)(x-3)`
`+B(x-3)+C(x-2)^(2)" "...(iii)`
On putting the values of `x=2,3` respectively, we get
`3xx2+2=B(2-3)`
`impliesB=-8`
`and 3xx3+2=(3-21)^(2)`
`impliesC=11`
On equating the coefficient of `x^(2)` in Eq. (iii), we get
`0=A+CimpliesA=-11`
On Putting the values of A, B and C in Eq. (ii), we get
`(3x+2)/((x-2)^(2)(x-3))=-(11)/((x-2))`
`-(8)/((x-2)^(2))+(11)/((x-3))`
`thereforel =int[-(11)/((x-2))-(8)/((x-2)^(2))+(11)/((x-3))]dx`
`=-11log(x-2)+(8)/((x-2))+11log (x-3)+c`
`=11log ((x-3)/(x-2))+(8)/((x-2))+c`