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The function $f$ whose graph passes through
$(0, 7 / 3)$ and whose derivative is $x \sqrt{1 - x^{2}}$ isgiven by

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$f (x) = (3 / 2) [\sin^{- 1} x + 6]$

$f (x) = - (2 / 3) [{(1 - x^{2})}^{3 / 2} - 8]$

$f (x) = - (1 / 3) [{(1 - x^{2})}^{3 / 2} - 8]$

$f (x) = (1 / 3) ({(1 - x^{2})}^{3 / 2} + 7)$

answer is C.

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Detailed Solution

$\begin{aligned} f (x) = \int x \sqrt{1 - x^{2}} d x \\ = - \frac{1}{2} \int (- 2 x) \sqrt{1 - x^{2}} d x \\ = - \frac{1}{3} {(1 - x^{2})}^{3 / 2} + C \end{aligned}$

So $7 / 3 = - 1 / 3 + C \Rightarrow C = 8 / 3$

Therefore $f (x) = - (1 / 3) [{(1 - x^{2})}^{3 / 2} - 8]$

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