Question

Evaluate:

${\int }_0^2\frac{dx}{x+4-x^2}$.

Answer 1

${\int }_0^2-\frac{dx}{x^2-\frac{2x}{2}+\frac{1}{4}-\frac{1}{4}-4}$

${\int }_0^2-\frac{dx}{(x-\frac{1}{2}{)}^2-\frac{17}{4}}$

${\int }_0^2\frac{1}{\sqrt{17}}(\frac{1}{x-\frac{1}{2}+\frac{\sqrt{17}}{2}}-\frac{1}{x-\frac{1}{2}-\frac{\sqrt{17}}{2}})dx$

$\frac{1}{\sqrt{17}}\log \frac{x-\frac{1}{2}+\frac{\sqrt{17}}{2}}{x-\frac{1}{2}-\frac{\sqrt{17}}{2}}$