Evaluate \(\int_{\frac{1}{2}}^{1} \frac{x^{3} - 4}{x^{3}} \mathrm {d} x\).

A.

-5.5

B.

-2.0

C.

2.0

D.

5.5

Correct answer is A

\(\int_{\frac{1}{2}}^{1} \frac{x^{3} - 4}{x^{3}} \mathrm {d} x \equiv \int_{\frac{1}{2}}^{1} 1 - \frac{4}{x^{3}} \mathrm {d} x\).

= \(\int_{\frac{1}{2}}^{1} 1 - 4x^{-3} \mathrm {d} x\).

= \(\left. x - \frac{4x^{-2}}{-2}\right |_{\frac{1}{2}}^{1} = \left. x + \frac{2}{x^{2}}\right |_{\frac{1}{2}}^{1}\)

= \((1 + 2) - (\frac{1}{2} + 8) = 3 - 8.5 = -5.5\)