Question

Evaluate:${\int }_{-1}^1[\sqrt{1+x+x^2}-\sqrt{1-x+x^2}]dx$

Answer 1

Let $I={\int }_{-1}^1[\sqrt{1+x+x^2}-\sqrt{1-x+x^2}]dx$
And $f\left(x\right)=[\sqrt{1+x+x^2}-\sqrt{1-x-x^2}]$
Then $f(-x)=[\sqrt{1+(-x)+{(-x)}^2}-\sqrt{1-(-x)-{(-x)}^2}]==[\sqrt{1-x+x^2}-\sqrt{1+x-x^2}]=-f\left(x\right)$
$\therefore I=0$