You have done this part:
#int(x^3+2x^2-3)/(x^2+x-1)dx= intxdx+intdx-2int1/(x^2+x-1)dx#
Quadratic formula:
#x = (-b+-sqrt(b^2-4(a)(c)))/(2a)#
#x = (-1+-sqrt(1^2-4(1)(-1)))/(2(1))#
#x = (-1+-sqrt(5))/2#
This makes the factors:
#(x+(1+sqrt5)/2)(x+(1-sqrt5)/2)#
Partial Fraction decomposition:
#1/((x+(1+sqrt5)/2)(x+(1-sqrt5)/2)) = A/(x+(1+sqrt5)/2)+ B/(x+(1-sqrt5)/2)#
Multiply by the denominator:
#1 = A(x+(1-sqrt5)/2)+ B(x+(1+sqrt5)/2)#
Eliminate B by letting #x = -(1+sqrt5)/2#
#1 = A(-(1+sqrt5)/2+(1-sqrt5)/2)#
#1 = A(-sqrt5)#
#A = -sqrt5/5#
Eliminate A by letting #x = -(1-sqrt5)/2#
#1 = B(-(1-sqrt5)/2+(1+sqrt5)/2)#
#1 = B(sqrt5)#
#B = sqrt5/5#
The integral becomes:
#int(x^3+2x^2-3)/(x^2+x-1)dx= intxdx+intdx-2sqrt5/5int1/(x+(1-sqrt5)/2)dx + 2sqrt5/5int1/(x+(1+sqrt5)/2)#
The integrals are trivial:
#int(x^3+2x^2-3)/(x^2+x-1)dx= x^2/2+x-2sqrt5/5ln|x+(1-sqrt5)/2| + 2sqrt5/5ln|x+(1+sqrt5)/2| + C#
