Question

\[\int\frac{x^5 + x^{- 2} + 2}{x^2} dx\]

Answer 1

\[\int\left( \frac{x^5 + x^{- 2} + 2}{x^2} \right)dx\]
\[ = \int \left( \frac{x^5}{x^2} + \frac{x^{- 2}}{x^2} + \frac{2}{x^2} \right)dx\]
\[ = \int\left( x^3 + x^{- 4} + 2 x^{- 2} \right)dx\]
\[ = \frac{x^{3 + 1}}{3 + 1} + \frac{x^{- 4 + 1}}{- 4 + 1} + 2\frac{x^{- 2 + 1}}{- 2 + 1} + C\]
\[ = \frac{x^4}{4} - \frac{1}{3 x^3} - \frac{2}{x} + C\]