11. The sequence $\left\{x_{n}\right\}$ is defined as follows: $x_{1}=\frac{1}{2}, x_{k+1}=x_{k}^{2}+x_{k}$. Then the integer part of the following sum $\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\cdots+\frac{1}{x_{100}+1}$ is . $\qquad$
11. From $x_{k+1}=x_{k}^{2}+x_{k} \Rightarrow \frac{1}{x_{k+1}}=\frac{1}{x_{k}\left(1+x_{k}\right)}=\frac{1}{x_{k}}-\frac{1}{1+x_{k}}$, we get
$$
\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\cdots+\frac{1}{x_{101}+1}=\frac{1}{x_{1}}-\frac{1}{x_{2}}+\cdots+\frac{1}{x_{100}}-\frac{1}{x_{101}}=\frac{1}{x_{1}}-\frac{1}{x_{101}}=2-\frac{1}{x_{101}},
$$

Since $x_{101}>1$, the answer should be 1.