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Consider the definite integral, $\displaystyle\int_{0}^{1}x\sqrt{1-x^{2}}dx$.

Using the substitution $u=1-x^{2}$, which of the following integrals is equivalent to the given integral?

A

$\cfrac{1}{2}\displaystyle\int_{0}^{1}\sqrt{u}du$

B

$-\cfrac{1}{2}\displaystyle\int_{0}^{1}\sqrt{u}du$

C

$\displaystyle\int_{0}^{1}\sqrt{1-u^{2}}du$

D

$\displaystyle\int_{0}^{1}\sqrt{u}du$

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