单选题 （共 5 题 ），每题只有一个选项正确
若反常积分 $\int_0^1 \frac{\ln x}{x^\alpha\left(\tan \frac{\pi}{2} x\right)^\beta} \mathrm{d} x$ 收敛, 则
$\text{A.}$ $-2 < \beta < 0$ 且 $\alpha+\beta \geqslant 1$.
$\text{B.}$ $0 < \beta < 2$ 且 $\alpha+\beta < 1$.
$\text{C.}$ $\beta < -2$ 且 $\alpha+\beta \geqslant 1$.
$\text{D.}$ $\beta>-2$ 且 $\alpha+\beta < 1$.
设 $I_1=\int_0^{\frac{\pi}{4}} \frac{\tan x}{x} \mathrm{~d} x, I_2=\int_0^{\frac{\pi}{4}}\left(\frac{\tan x}{x}\right)^2 \mathrm{~d} x, I_3=\int_0^{\frac{\pi}{4}} \frac{\tan x^2}{x^2} \mathrm{~d} x$, 则有
$\text{A.}$ $I_1>I_2>I_3$
$\text{B.}$ $I_3>I_2>I_1$
$\text{C.}$ $I_2>I_1>I_3$
$\text{D.}$ $I_1>I_3>I_2$
计算积分 $\int_0^{\frac{\pi}{4}} \mathrm{~d} \theta \int_0^{\tan \theta \sec \theta} \mathrm{e}^{r \cos \theta} r \mathrm{~d} r+\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \mathrm{~d} \theta \int_0^{\frac{1}{\sin \theta}} \mathrm{e}^{r \cos \theta} r \mathrm{~d} r=$.
$\text{A.}$ e
$\text{B.}$ ${e}^{-1}$
$\text{C.}$ 0
$\text{D.}$ 1
当 $0 < x \leqslant \frac{\pi}{4}$ 时,下列不等式成立的是
$\text{A.}$ $\int_0^x \mathrm{e}^t \cos t \mathrm{~d} t>x, \int_0^x \mathrm{e}^t(1-\sin t) \mathrm{d} t < x$.
$\text{B.}$ $\int_0^x \mathrm{e}^t \cos t \mathrm{~d} t < x, \int_0^x \mathrm{e}^t(1-\sin t) \mathrm{d} t>x$.
$\text{C.}$ $\int_0^x \mathrm{e}^t \cos t \mathrm{~d} t>x, \int_0^x \mathrm{e}^t(1-\sin t) \mathrm{d} t>x$.
$\text{D.}$ $\int_0^x \mathrm{e}^t \cos t \mathrm{~d} t < x, \int_0^x \mathrm{e}^t(1-\sin t) \mathrm{d} t < x$.
下列反常积分中, 收敛的是
$\text{A.}$ $\int_0^{+\infty} \frac{1}{\sqrt{x^2+x}} \mathrm{~d} x$.
$\text{B.}$ $\int_0^{+\infty} \frac{1}{\sqrt{x^3(x+1)^3}} \mathrm{~d} x$.
$\text{C.}$ $\int_0^{+\infty} \frac{1}{\sqrt[3]{x^3+x^2}} \mathrm{~d} x$.
$\text{D.}$ $\int_0^{+\infty} \frac{1}{\sqrt{x^3+x}} \mathrm{~d} x$.
填空题 （共 1 题 ），请把答案直接填写在答题纸上
已知微分方程 $y^{\prime}-x \sin 2 y=\frac{\ln x}{\sqrt{\left(1+x^2\right)^3}} \cos ^2 y$, 则不定积分 $\int x \tan y \mathrm{~d} x=$