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,cto 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