Find \(\int \frac{x^{3} + 5x + 1}{x^{3}} \mathrm {d} x\)
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Correct Answer: Option D
Explanation:
\(\frac{x^{3} + 5x + 1}{x^{3}} \equiv 1 + \frac{5}{x^{2}} + \frac{1}{x^{3}}\)
\(\equiv \int (1 + \frac{5}{x^{2}} + \frac{1}{x^{3}}) \mathrm {d} x = \int (1 + 5x^{-2} + x^{-3}) \mathrm {d} x\)
= \((x - 5x^{-1} - \frac{1}{2}x^{-2} + c)\)
= \(x - \frac{5}{x} - \frac{1}{2x^{2}} + c\).
\(\frac{x^{3} + 5x + 1}{x^{3}} \equiv 1 + \frac{5}{x^{2}} + \frac{1}{x^{3}}\)
\(\equiv \int (1 + \frac{5}{x^{2}} + \frac{1}{x^{3}}) \mathrm {d} x = \int (1 + 5x^{-2} + x^{-3}) \mathrm {d} x\)
= \((x - 5x^{-1} - \frac{1}{2}x^{-2} + c)\)
= \(x - \frac{5}{x} - \frac{1}{2x^{2}} + c\).