Consider the expression
(12x^(2)+13x+2)/(3x-2)
. The numerator is a polynomial of degree 2 , and the denominator is a polynomial of degree 1. Because the degree of the numerator is greater than the degree of the denominator, we can use long division to find real numbers
a,b,c
to write
(12x^(2)+13x+2)/(3x-2)= polynomial + remainder
=(ax+b)+(c)/(3x-2)
Using long division:
(12x^(2)+13x+2)/(3x-2)=
x++,(1)/(3x-2)
Therefore
\int (12x^(2)+13x+2)/(3x-2)dx=
+,+,+C