Question

$\int \frac{1+x+x^2}{x^2\left(1+x\right)}dx$

Answer 1

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