# What is int (-2x^3-x^2+6x+9 ) / (2x^2- x +3 )?

Nov 18, 2017

$\int \frac{\left(- 2 {x}^{3} - {x}^{2} + 6 x + 9\right) \cdot \mathrm{dx}}{2 {x}^{2} - x + 3}$

=$2 L n \left(2 {x}^{2} - x + 3\right) + \frac{20 \sqrt{23}}{23} \cdot \arctan \left[\frac{4 x - 1}{\sqrt{23}}\right] - {x}^{2} / 2 - x + C$

#### Explanation:

$\int \frac{\left(- 2 {x}^{3} - {x}^{2} + 6 x + 9\right) \cdot \mathrm{dx}}{2 {x}^{2} - x + 3}$

$= - \int \frac{\left(2 {x}^{3} + {x}^{2} - 6 x - 9\right) \cdot \mathrm{dx}}{2 {x}^{2} - x + 3}$

=-$\int \left(x + 1\right) \cdot \mathrm{dx} - \int \frac{\left(- 8 x - 12\right) \cdot \mathrm{dx}}{2 {x}^{2} - x + 3}$

=$- \left({x}^{2} / 2 + x\right) + C + \int \frac{\left(8 x + 12\right) \cdot \mathrm{dx}}{2 {x}^{2} - x + 3}$

=$- {x}^{2} / 2 - x + C + 2 \cdot \int \frac{\left(4 x + 1\right) \cdot \mathrm{dx}}{2 {x}^{2} - x + 3} + \int \frac{10 \cdot \mathrm{dx}}{2 {x}^{2} - x + 3}$

=$- {x}^{2} / 2 - x + C + 2 L n \left(2 {x}^{2} - x + 3\right) + \int \frac{80 \cdot \mathrm{dx}}{16 {x}^{2} - 8 x + 24}$

=$2 L n \left(2 {x}^{2} - x + 3\right) - {x}^{2} / 2 - x + C + 20 \cdot \int \frac{4 \cdot \mathrm{dx}}{{\left(4 x - 1\right)}^{2} + 23}$

=$2 L n \left(2 {x}^{2} - x + 3\right) - {x}^{2} / 2 - x + C + \frac{20 \sqrt{23}}{23} \cdot \arctan \left[\frac{4 x - 1}{\sqrt{23}}\right]$

=$2 L n \left(2 {x}^{2} - x + 3\right) + \frac{20 \sqrt{23}}{23} \cdot \arctan \left[\frac{4 x - 1}{\sqrt{23}}\right] - {x}^{2} / 2 - x + C$