Evaluate `int((2x+1))/((4-3x-x^(2)))dx`.

Explanation

Let `(2x+1)=A*(d)/(dx)(4-3x-x^(2))+B`.
Then , `(2x+1)=A(-3-2x)+B` . . . (i)
Comparing the coefficients of like terms , we get
`(-2A=2and-3A+B=1)rArr(A=-1,B=-2)`.
`thereforeint((2x+1))/((4-3x-x^(2)))dx = int{((-1)*(-3-2x)-2)/((4-3x-x^(2)))}dx`
`=-int((3-2x))/((4-3x-x^(2)))dx-2int(dx)/((4-3x-x^(2)))`
`=-log|4-3x-x^(2)|+2int(dx)/((4-3x-x^(2)))`
`=-log|4-3x^(2)-x^(2)|+2int(dx)/((x+(2)/(2))^(2)-(4+(9)/(4)))`
`=-log|4-3x-x^(2)|+2int(dx)/({(x+(3)/(2))^(2)-((5)/(2))^(2)})`
`=-log|4-3x-x^(2)|+(2)/((2xx(5)/(2)))log|((x+(3)/(2))-(5)/(2))/((x+(3)/(2))+(5)/(2))|+C`
`=-log|4-3x-x^(2)|+(2)/(5)log|(x-1)/(x+4)|+C`.