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The easiest way to work this multiple-choice problem is by working backward. We are given the function $f(x) = 3\left(1 + x^{-1/2}\right)^{-k/3} + C\ \ \ k = 1, 2$ and differentiate: $f'(x) = 3\cdot \frac{-k}{3}\left(1 + x^{-1/2}\right)^{-(k+3)/3}\cdot \frac{-1}{2}x^{-3/2}$ $= \frac k 2 \left(1 + x^{-1/2}\right)^{-(k+3)/3}x^{-3/2}.$ Comparing with the given integrand, we see that $k + 3 = 5$ , so $k = 2$ ; but there are some descrepancies. Note, however, that $x^{-3/2}\left(1 + x^{-1/2}\right)^{-5/3} = x^{-2/3}\cdot x^{-5/6}\cdot \left(1 + x^{-1/2}\right)^{-5/3} = x^{-2/3}\cdot \left(\left(1 + x^{-1/2}\right)\cdot x^{1/2}\right)^{-5/3} = x^{-2/3}\cdot \left(x^{1/2} + 1\right)^{-5/3},$ so that we have indeed the correct function.

Since $k = 2$ the answer is $\boxed{\mathbf A}$ .