Answer
$$
\int \frac{x+1}{x^{2}+2 x} d x =\frac{1}{2} \ln \left|x^{2}+2 x\right|+C
$$
where $C$ is an arbitrary constant.
Work Step by Step
$$
\int \frac{x+1}{x^{2}+2 x} d x
$$
$$\text { Let } u=x^{2}+2 x . \text { Then } d u=(2 x+2) d x=2(x+1) d x
$$
substituting in the given integral we have :
$$
\begin{aligned}
\int \frac{x+1}{x^{2}+2 x} d x& =\int \frac{\frac{1}{2} d u}{u} \\
&=\frac{1}{2} \ln |u|+C \\
&=\frac{1}{2} \ln \left|x^{2}+2 x\right|+C
\end{aligned}
$$
where $C$ is an arbitrary constant.