<s>[INST] How to solve $\int \frac{dx}{x+x^2}$ without using partial fractions? I can easily rewrite it as $\frac{1}{x} - \frac{1}{x+1}$, but is it possible to solve this without partial fractions?
 [/INST] Let $\displaystyle y=1+\frac{1}{x}$. Then $\displaystyle \frac{dy}{dx}=-\frac{1}{x^2}$.
\begin{align*}
\int\frac{dx}{x+x^2}&=-\int\frac{\frac{-1}{x^2}dx}{1+\frac{1}{x}}\\
&=-\ln\left(1+\frac{1}{x}\right)+C
\end{align*}
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