Question

The value of \[\int_{-1}^{1}\left(\sqrt{1+x+x^{2}}-\sqrt{1-x+x^{2}}\right)\mathrm{d}x\] is

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Answer 1

0

Explanation:

Let f (x) = \[\sqrt{1+x+x^{2}}-\sqrt{1-x+x^{2}}\]

\[\mathrm{f}(-x)=\sqrt{1-x+x^2}-\sqrt{1+x+x^2}\]

\[=-\left(\sqrt{1+x+x^2}-\sqrt{1-x+x^2}\right)\]

= – f (x)

∴ The given function is odd.

\[\therefore\quad\int_{-1}^1\left(\sqrt{1+x+x^2}-\sqrt{1-x+x^2}\right)\mathrm{~d}x=0\]