解答题 (共 17 题 ),解答过程应写出必要的文字说明、证明过程或演算步骤
$\lim _{n \rightarrow \infty} \frac{n^{n+1}}{(n+1)^n} \sin \frac{1}{n}$
$\lim _{n \rightarrow \infty}\left(\frac{\sqrt[n]{a}+\sqrt[n]{b}+\sqrt[n]{c}}{3}\right)^n$ 其中 $a>0, b>0, c>0$ .
$\lim _{x \rightarrow 0} \frac{(1-\cos x)[x-\ln (1+\tan x)]}{\sin ^4 x}$.
$\lim _{x \rightarrow 0} \frac{\mathrm{e}-\mathrm{e}^{\cos x}}{\sqrt[3]{1+x^2}-1}$
设 $f(x)$ 二阶可导, $f(0)=0, f^{\prime}(0)=1, f^{\prime \prime}(0)=2$. 求极限 $\lim _{x \rightarrow 0} \frac{f(x)-x}{x^2}$.
$\lim _{n \rightarrow \infty}\left(\frac{1}{n^2+n+1}+\frac{2}{n^2+n+2}+\cdots+\frac{n}{n^2+n+n}\right)$
设 $x_1>0, x_{n+1}=\frac{1}{2}\left(x_n+\frac{1}{x_n}\right), n=1,2, \cdots$. 求极限 $\lim _{n \rightarrow \infty} x_n$.
求极限 $\lim _{n \rightarrow \infty}\left(\frac{1}{n^2+1^2}+\frac{2}{n^2+2^2}+\cdots+\frac{n}{n^2+n^2}\right)$.
计算 $\int_1^{+\infty} \frac{\mathrm{d} x}{\mathrm{e}^x+\mathrm{e}^{2-x}}$
设 $f(x)=\int_0^x \frac{\sin t}{\pi-t} \mathrm{~d} t $ 计算 $ \int_0^\pi f(x) \mathrm{d} x $
$ \int \frac{d x}{\sin ^6 x+\cos ^6 x}$
$\int e^{\frac{x}{2}} \frac{\cos x}{\sqrt{\sin x+\cos x}} d x$
$\int e^{2 x}(\tan x+1)^2 d x$
$\int(\sqrt{\cot x}-\sqrt{\tan x}) d x$
$\int \frac{x \ln x}{\left(1+x^2\right)^{\frac{3}{2}}} d x$
$\int \frac{\sin (\ln x)}{x^2} d x$
$\int \frac{\sin 2 x}{\sin x+\cos x} d x$