单选题 （共 3 题 ），每题只有一个选项正确
$\lim _{n \rightarrow \infty} \ln \sqrt[n]{\left(1+\frac{1}{n}\right)^2\left(1+\frac{2}{n}\right)^2 \cdots\left(1+\frac{n}{n}\right)^2}$ 等于
$\text{A.}$ $\int_1^2 \ln ^2 x \mathrm{~d} x$.
$\text{B.}$ $2 \int_1^2 \ln x \mathrm{~d} x$.
$\text{C.}$ $2 \int_1^2 \ln (1+x) \mathrm{d} x$.
$\text{D.}$ $\int_1^2 \ln ^2(1+x) \mathrm{d} x$.
设 $M=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{(1+x)^2}{1+x^2} \mathrm{~d} x, N=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{1+x}{\mathrm{e}^x} \mathrm{~d} x, K=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}(1+\sqrt{\cos x}) \mathrm{d} x$, 则
$\text{A.}$ $M>N>K$.
$\text{B.}$ $M>K>N$.
$\text{C.}$ $K>M>N$.
$\text{D.}$ $K>N>M$.
设 $a_n=\frac{3}{2} \int_0^{\frac{n}{n+1}} x^{n-1} \sqrt{1+x^n} \mathrm{~d} x$ ，则极限 $\lim _{n \rightarrow \infty} n a_n$ 等于
$\text{A.}$ $(1+\mathrm{e})^{\frac{3}{2}}+1$ ．
$\text{B.}$ $\left(1+\mathrm{e}^{-1}\right)^{\frac{3}{2}}-1$ ．
$\text{C.}$ $\left(1+\mathrm{e}^{-1}\right)^{\frac{3}{2}}+1$ ．
$\text{D.}$ $(1+\mathrm{e})^{\frac{3}{2}}-1$ ．