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$$
\int_{-1}^{1}\left[\sqrt{1+x+x^{2}}-\sqrt{1-x+x^{2}}\right] d x=
$$
Options:
\int_{-1}^{1}\left[\sqrt{1+x+x^{2}}-\sqrt{1-x+x^{2}}\right] d x=
$$
Solution:
2213 Upvotes
Verified Answer
The correct answer is:
0
Given I $=\int_{-1}^{1}\left(\sqrt{1+x+x^{2}}-\sqrt{1-x+x^{2}}\right) d x$
Let $f(x)=\sqrt{1+x+x^{2}}-\sqrt{1-x+x^{2}}$
$\therefore f(-x)=\sqrt{1-x+x^{2}}-\sqrt{1+x+x^{2}}=-\left(\sqrt{1+x+x^{2}}-\sqrt{1-x-x^{2}}\right)=-f(x)$
$\therefore I=0$
Let $f(x)=\sqrt{1+x+x^{2}}-\sqrt{1-x+x^{2}}$
$\therefore f(-x)=\sqrt{1-x+x^{2}}-\sqrt{1+x+x^{2}}=-\left(\sqrt{1+x+x^{2}}-\sqrt{1-x-x^{2}}\right)=-f(x)$
$\therefore I=0$
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