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