Question

\[\int\frac{1 + x + x^2}{x^2 \left( 1 + x \right)} \text{ dx}\]

Answer 1

\[\int\frac{1 + x + x^2}{x^2 \left( 1 + x \right)}dx\]
\[ \Rightarrow \int\frac{\left( 1 + x \right)}{x^2 \left( 1 + x \right)}dx + \int\frac{x^2}{x^2 \left( 1 + x \right)}dx\]
\[ \Rightarrow \int\frac{dx}{x^2} + \int\frac{dx}{1 + x}\]
\[ \Rightarrow \int x^{- 2} dx + \int\frac{1}{1 + x}dx\]
\[ \Rightarrow \left[ \frac{x^{- 2 + 1}}{- 2 + 1} \right] + \text{ ln }\left( 1 + x \right) + C\]
\[ \Rightarrow \frac{- 1}{x} + \text{ ln } \left| 1 + x \right| + C\]