Do partial fraction decomposition:
(2x - 3)/((x - 2)(x - 3)) = A/(x - 2) + B/(x - 3)2x−3(x−2)(x−3)=Ax−2+Bx−3
Multiply both sides by ((x - 2)(x - 3))((x−2)(x−3))
(2x - 3) = A(x - 3) + B(x - 2)(2x−3)=A(x−3)+B(x−2)
Solve for A by letting x = 2x=2:
(2(2) - 3) = A(2 - 3)(2(2)−3)=A(2−3)
-1 = A−1=A
Solve for B by letting x = 3x=3:
(2(3) - 3) = B(3 - 2)(2(3)−3)=B(3−2)
3 = B
Check:
-1/(x - 2) + 3/(x - 3) = −1x−2+3x−3=
-1/(x - 2)(x - 3)/(x - 3) + 3/(x - 3)(x - 2)/(x - 2) =−1x−2x−3x−3+3x−3x−2x−2=
(-x + 3)/((x - 2)(x - 3)) + (3x - 6)/((x - 3)(x - 2)) =−x+3(x−2)(x−3)+3x−6(x−3)(x−2)=
(2x - 3)/((x - 3)(x - 2))2x−3(x−3)(x−2)
This checks.
int (2x - 3)/(x^2 - 5x + 6)dx = 3int 1/(x - 3)dx - int1/(x - 2)dx∫2x−3x2−5x+6dx=3∫1x−3dx−∫1x−2dx
int (2x - 3)/(x^2 - 5x + 6)dx = 3ln|x - 3| - ln|x - 2| + C∫2x−3x2−5x+6dx=3ln|x−3|−ln|x−2|+C
