3．$\underset{x\to +\infty }{\mathop{\lim }}\,\frac{{{\text{e}}^{-{{x}^{2}}}}}{x}\int_{0}^{x}{{{t}^{2}}{{\text{e}}^{{{t}^{2}}}}\,\text{d}t}=$()．
A. $\text{e}$
B. $\frac{\text{e}}{2}$
C. $1$
D. $\frac{1}{2}$
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4．设函数$f(x)$在区间$[0,\ 1]$上连续，且满足$f(0)=1,\ f(1)=2$，则$\underset{n\to \infty }{\mathop{\lim }}\,\int_{0}^{1}{{{x}^{n}}f(x)\text{d}x}$()．
A. 等于$2$
B. 等于$1$
C. 等于$0$
D. 不存在
5．$\int_{\,0}^{\,\,1}{\frac{\text{d}x}{{{\text{e}}^{x}}+{{\text{e}}^{-x}}}=}$()．
A. $\arctan \text{e}+\frac{\text{ }\!\!\pi\!\!\text{ }}{2}$
B. $\arctan \text{e}-\frac{\text{ }\!\!\pi\!\!\text{ }}{2}$
C. $\arctan \text{e}+\frac{\text{ }\!\!\pi\!\!\text{ }}{4}$
D. $\arctan \text{e}-\frac{\text{ }\!\!\pi\!\!\text{ }}{4}$
6．$\int_{0}^{1}{\frac{\arcsin \sqrt{x}}{\sqrt{x(1-x)}}\,\,\text{d}x}$()．
A. ${{\left( \frac{\text{ }\!\!\pi\!\!\text{ }}{2} \right)}^{2}}$
B. $\frac{\text{ }\!\!\pi\!\!\text{ }}{2}$
C. ${{\left( \frac{\text{ }\!\!\pi\!\!\text{ }}{4} \right)}^{2}}$
D. $\frac{\text{ }\!\!\pi\!\!\text{ }}{4}$
4．设$S(x)=\sum\limits_{n=1}^{\infty }{{{(-1)}^{n-1}}n(n+1){{x}^{n}}}$，$x\in (-1,\ 1)$，则$S(\frac{1}{2})=$()．
A. $\frac{18}{64}$
B. $\frac{27}{64}$
C. $\frac{4}{27}$
D. $\frac{8}{27}$
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