Question Detail

First, factor the denominator: x² - x - 6 = (x - 3)(x + 2).Use partial fraction decomposition:(x+1) / ((x-3)(x+2)) = A/(x-3) + B/(x+2).Multiply both sides by (x-3)(x+2): x+1 = A(x+2) + B(x-3).To find A, set x = 3: 3+1 = A(3+2) => 4 = 5A => A = 4/5.To find B, set x = -2: -2+1 = B(-2-3) => -1 = -5B => B = 1/5.So the integral becomes: ∫[ (4/5)/(x-3) + (1/5)/(x+2) ] dx.= (4/5)∫(1/(x-3)) dx + (1/5)∫(1/(x+2)) dx.= (4/5)ln|x-3| + (1/5)ln|x+2| + C.Using logarithm properties (a ln b = ln ba and ln a + ln b = ln(ab)):= (1/5)[ln|(x-3)4| + ln|x+2|] + C.= (1/5)ln[|(x-3)4(x+2)|] + C.This can be written as ln[5√((x-3)4(x+2))] + C, assuming the domain allows for positive arguments in the logarithm.