Question

What is `int (-2x^3-x^2+x+2 ) / (3x^2+2x +3 )`?

Answer 1

`(25Ln(3x^2+2x+3)+10sqrt2arctan((3x+1)/(2sqrt2))-18x^2+6x)/54+C` or

=`25/54Ln(3x^2+2x+3)+(5sqrt2)/27arctan((3x+1)/(2sqrt2))-(3x^2-x)/9+C`

Explanation:

`int (-2x^3-x^2+x+2)/(3x^2+2x+3)*dx`

=`-1/9int (18x^3+9x^2-9x-18)/(3x^2+2x+3)*dx`

=`-1/9int ((6x-1)*(3x^2+2x+3)-(25x+15))/(3x^2+2x+3)*dx`

=`-1/9int (6x-1)dx`+`1/9int (25x+15)/(3x^2+2x+3)*dx`

=`-1/9*(3x^2-x)+C`+`1/9int (25/6*(6x+2)+20/3)/(3x^2+2x+3)*dx`

=`-1/9*(3x^2-x)+C`+`25/54int (6x+2)/(3x^2+2x+3)*dx`+`20/27int (dx)/(3x^2+2x+3)`

=`-1/9*(3x^2-x)+25/54ln(3x^2+2x+3)+C`+`20/27int (3dx)/(9x^2+6x+9)`

=`-1/9*(3x^2-x)+25/54ln(3x^2+2x+3)+C`+`20/27int (3dx)/((3x+1)^2+(2sqrt2)^2)`

=`-1/9*(3x^2-x)+25/54ln(3x^2+2x+3)`+`(5sqrt2)/27arctan((3x+1)/(2sqrt2))+C`

=`(25Ln(3x^2+2x+3)+10sqrt2arctan((3x+1)/(2sqrt2))-18x^2+6x)/54+C`