`int (2x-3)/(x^(2)+3x-18) dx`

Options

A. `1/3 log |x-3|+5/3 log |x+6|+c`
B. ` log |x-3|+5/3 log |x+6|+c`
C. `1/3 log |x-3|+log |x+6|+c`
D. `log |x-3|+ log |x+6|+c`

Answer

A

Explanation

`(2x-3)/(x^(2)+3x-18)=(2x-3)/((x-3)(x+6))`
`=A/(x-3)+B/(x+6)`
या `2x-3=A(x+6)+B(x-3)`
जब `x-3=0` या `x=3`
`3=9A` या `A=1/3`
जब `x+6=0` या `x=-6`
`-15=B(-9)` या `B=5/3`
तब `int (2x-3)/(x^(2)+3x-18) dx =1/3 int (dx)/(x-3)+5/3 int (dx)/(x+6)`
`=1/3 log |x-3|+5/3 log |x+6|+c`.