Question Detail

First, factor the denominator: x² - x - 6 = (x - 3)(x + 2).Now, 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.Now, integrate the decomposed fractions:∫ [ (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 b^a and ln a + ln b = ln(ab)):= (1/5) [ 4 ln|x-3| + ln|x+2| ] + C= (1/5) [ ln((x-3)⁴) + ln(x+2) ] + C= (1/5) ln[ (x-3)⁴(x+2) ] + C= ln[ ((x-3)⁴(x+2))^(1/5) ] + C= ln⁵√((x-3)⁴(x+2)).