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: A
Explanation
\(\int_{\frac{1}{2}}^{1} \frac{x^{3} - 4}{x^{3}} \mathrm {d} x \equiv \int_{\frac{1}{2}}^{1} 1 - \frac{4}{x^{3}} \mathrm {d} x\).
= \(\int_{\frac{1}{2}}^{1} 1 - 4x^{-3} \mathrm {d} x\).
= \(\left. x - \frac{4x^{-2}}{-2}\right |_{\frac{1}{2}}^{1} = \left. x + \frac{2}{x^{2}}\right |_{\frac{1}{2}}^{1}\)
= \((1 + 2) - (\frac{1}{2} + 8) = 3 - 8.5 = -5.5\)
