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$$ \int_{-\sqrt{\vphantom{\large A}2\,}\,}^{\sqrt{\vphantom{\large A}2\,}\,} x\sqrt{4 - x^{4}}\,{\rm d}x $$ I find the answer $0$ by substitution $x^{2} = u$, but it didn't make sense.

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up vote 12 down vote accepted

No substitution is needed. The integrand is an odd function of $x$ and you are integrating on an interval that is symmetric about $0$. Therefore the integral is $0$.

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make sense with this substitution x=2 sin a

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