int_{-1}^1 left(sqrt{1+x+x^2}-sqrt{1-x+x^2}ri | vedclass

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(C) Let $I = \int_{-1}^1 \left(\sqrt{1+x+x^2}-\sqrt{1-x+x^2}\right) dx$. Define $f(x) = \sqrt{1+x+x^2} - \sqrt{1-x+x^2}$. We check if $f(x)$ is an odd

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(C) Let $I = \int_{-1}^1 \left(\sqrt{1+x+x^2}-\sqrt{1-x+x^2}\right) dx$. Define $f(x) = \sqrt{1+x+x^2} - \sqrt{1-x+x^2}$. We check if $f(x)$ is an odd function: $f(-x) = \sqrt{1+(-x)+(-x)^2} - \sqrt{1-(-x)+(-x)^2} = \sqrt{1-x+x^2} - \sqrt{1+x+x^2}$. Since $f(-x) = -(\sqrt{1+x+x^2} - \sqrt{1-x+x^2}) = -f(x)$,the function $f(x)$ is an odd function. For an odd function,$\int_{-a}^a f(x) dx = 0$. Therefore,$\int_{-1}^1 \left(\sqrt{1+x+x^2}-\sqrt{1-x+x^2}\right) dx = 0$.

$\int_{-1}^1 \left(\sqrt{1+x+x^2}-\sqrt{1-x+x^2}\right) dx =$

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