Question

evaluate the intergration of (2-5X-x2)dx over (x^2-x-6)(x+2)

Answer 1

$$\text{Evaluate } \int \frac {2 - 5 x - x ^ {2}}{x ^ {2} - x - 6} \, dx$$ $$\int \frac {2 - 5 x - x ^ {2}}{x ^ {2} - x - 6} \, dx = - \int \frac {x ^ {2} - x - 6 + 6 x + 4}{x ^ {2} - x - 6} \, dx = - \int dx - \int \frac {6 x + 4}{(x + 2) (x - 3)} \, dx$$ $$\frac {6 x + 4}{(x + 2) (x - 3)} = \frac {A}{(x + 2)} + \frac {B}{(x - 3)}$$ $$A x - 3 A + B x + 2 B = 6 x + 4$$ $$\left\{ \begin{array}{c} A + B = 6 \\ - 3 A + 2 B = 4 \end{array} \right.$$ $$A = \frac {8}{5}$$ $$B = \frac {22}{5}$$ $$\int \frac {2 - 5 x - x ^ {2}}{x ^ {2} - x - 6} \, dx = - x - \frac {8}{5} \int \frac {dx}{(x + 2)} - \frac {22}{5} \int \frac {dx}{(x - 3)} = - x - \frac {8}{5} \ln (x + 2) - \frac {22}{5} \ln (x - 3) + C$$