Evaluate the integral: int_{-1}^{1} left[ sqr | vedclass

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(D) Let $I = \int_{-1}^{1} \left( \sqrt{1+x+x^{2}} - \sqrt{1-x+x^{2}} \right) dx$. Define $f(x) = \sqrt{1+x+x^{2}} - \sqrt{1-x+x^{2}}$. Now,calculate

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(D) Let $I = \int_{-1}^{1} \left( \sqrt{1+x+x^{2}} - \sqrt{1-x+x^{2}} \right) dx$. Define $f(x) = \sqrt{1+x+x^{2}} - \sqrt{1-x+x^{2}}$. Now,calculate $f(-x)$: $f(-x) = \sqrt{1+(-x)+(-x)^{2}} - \sqrt{1-(-x)+(-x)^{2}} = \sqrt{1-x+x^{2}} - \sqrt{1+x+x^{2}}$. This can be written as $f(-x) = - \left( \sqrt{1+x+x^{2}} - \sqrt{1-x+x^{2}} \right) = -f(x)$. Since $f(x)$ is an odd function,by the property of definite integrals $\int_{-a}^{a} f(x) dx = 0$ if $f(x)$ is odd,we have $I = 0$.

Evaluate the integral: $\int_{-1}^{1} \left[ \sqrt{1+x+x^{2}} - \sqrt{1-x+x^{2}} \right] dx$

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