How to solve $\int \frac{dx}{x+x^2}$ without using partial fractions?

Question

I can easily rewrite it as $\frac{1}{x} - \frac{1}{x+1}$, but is it possible to solve this without partial fractions?

Answer 1

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*}