Let $f$ be a real-valued function defined on the interval $\left(0,\infty \right)$ by $f\left(x\right)=\ln x+{\int }_0^x\sqrt{1+\sin t}dt$, then which of the following statement(s) is (are) true?

For any real number $x$, let $\left[x\right]$ denote the largest integer less than or equal to $x$. Let $f$ be a real-valued function defined on the interval $\left[-10,10\right]$ by  

$f\left(x\right)=\left\{\begin{array}{cc}x-\left[x\right],& \text{if} \left[x\right] \text{is odd}\\ 1+\left[x\right]-x,& \text{if} \left[x\right] \text{is even}\end{array}\right.$

Then, the value of $\frac{{\mathrm{\pi }}^2}{10}{\int }_{-10}^{10}f\left(x\right) \mathrm{cos\pi }xdx$ is

Let $f:\left[\frac{1}{2},1\right]\to R$ (the set of all real numbers) be a positive, non-constant, and differentiable function such that $f^{\text{'}}\left(x\right)<2f\left(x\right)$ and $f\left(\frac{1}{2}\right)=1$. Then the value of ${\int }_{\frac{1}{2}}^{1}f\left(x\right)dx$ lies in the interval,