How do you find the integral, $\int2x\,dx/(1+x)$ Integral: $\displaystyle \int \dfrac{2x}{1+x}dx$. I think I have to use $u$ substitution, but I'm having trouble understanding what to do. Thank you in advance!

Here's a trick against long division:
$$
\begin{align}
\int \frac{2x}{1 + x}\mathrm dx &= 2\int \frac{(x + 1) - 1}{(x + 1)}\mathrm dx \\
&= 2\left(\int\mathrm dx -\int\frac{\mathrm d(x+1)}{x+1}\right)\\
&= 2\Big(x - \ln|x+1|\Big) \color{grey}{+ \mathcal C}
\end{align}$$
Note that $2x + \ln\left|\frac{1}{x^2 + 2x + 1}\right| + \color{grey}{\mathcal C}$ is also a valid answer =)
