Integrate (x^2+4x+13)/((x+2)^2)(x-1) dx by using partial fractions
Express (x2+4x+13) / (x+2)2(x-1) as partial fractions. (x2+4x+13) / (x+2)2(x-1) = a/(x+2) +b/(x+2)2 +c/(x-1) where a, b and c are constants to be found. Multiplying by the denominator, we get (x2+4x+13) = a(x+2)(x-1) + b(x-1) + c(x+2)2 By setting x=1, we get 18=9c so c=2 By setting x=-2, we get 9=-3b so b=-3 By setting x=0 (or any other number) and using c=2 and b=-3, we get (for x=0) 13=-2a+3+8 so a=-1 Hence, (x2+4x+13) / (x+2)2(x-1) = -1/(x+2) -3/(x+2)2 + 2/(x-1) Integrating the partial fraction, we get -1ln(x+2) + (-3)(-1)(x+2)-2+1 + 2ln(x-1) +c where c is the constant of integration This simplifies down to -ln(x+2) +3/(x+2) +2ln(x-1) +c
