单选题 （共 6 题 ），每题只有一个选项正确
设 $M=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\sin x}{1+x^{2}} \cos ^{4} x \mathrm{~d} x, N=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\sin ^{3} x+\cos ^{4} x\right) \mathrm{d} x, P=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(x^{2} \sin ^{3} x-\cos ^{4} x\right) \mathrm{d} x$, 则有
$\text{A.}$ $N < P < M$.
$\text{B.}$ $M < P < N$.
$\text{C.}$ $N < M < P$.
$\text{D.}$ $P < M < N$.
下列广义积分收敛的是
$\text{A.}$ $\int_e^{+\infty} \frac{\ln x}{x} \mathrm{~d} x$
$\text{B.}$ $\int_e^{+\infty} \frac{\mathrm{d} x}{x \ln x}$
$\text{C.}$ $\int_e^{+\infty} \frac{\mathrm{d} x}{x(\ln x)^2}$
$\text{D.}$ $\int_e^{+\infty} \frac{\mathrm{d} x}{x \sqrt{\ln x}}$
设函数 $f(x)$ 在 $(-\infty,+\infty)$ 上连续，则 $\mathrm{d}\left[\int f(x) \mathrm{d} x\right]$ 等于
$\text{A.}$ $f(x)$
$\text{B.}$ $f(x) \mathrm{d} x$
$\text{C.}$ $f(x)+C$
$\text{D.}$ $f^{\prime}(x) \mathrm{d} x$
设函数 $f(x)=\left\{\begin{array}{cc}x^2 & 0 \leq x \leq 1 \\ 2-x & 1 < x \leq 2\end{array}\right.$ ，记$F(x)=\int_0^x f(t) \mathrm{d} t, 0 \leq x \leq 2$, 则
$\text{A.}$ $F(x)=\left\{\begin{array}{cl}\frac{x^3}{3}, & 0 \leq x \leq 1 \\ \frac{1}{3}+2 x-\frac{x^2}{2}, & 1 < x \leq 2\end{array}\right.$
$\text{B.}$ $F(x)=\left\{\begin{array}{cc}\frac{x^3}{3}, & 0 \leq x \leq 1 \\ -\frac{7}{6}+2 x-\frac{x^2}{2}, 1 < x \leq 2\end{array}\right.$
$\text{C.}$ $F(x)=\left\{\begin{array}{cr}\frac{x^3}{3}, & 0 \leq x \leq 1 \\ \frac{x^3}{3}+2 x-\frac{x^2}{2}, 1 < x \leq 2\end{array}\right.$
$\text{D.}$ $F(x)=\left\{\begin{array}{c}\frac{x^3}{3}, \quad 0 \leq x \leq 1 \\ 2 x-\frac{x^2}{2}, 1 < x \leq 2\end{array}\right.$
设 $f(x)$ 连续， $F(x)=\int_0^{x^2} f\left(t^2\right) \mathrm{d} t$ ，则 $F^{\prime}(x)$ 等于
$\text{A.}$ $f\left(x^4\right)$
$\text{B.}$ $x^2 f\left(x^4\right)$
$\text{C.}$ $2 x f\left(x^4\right)$
$\text{D.}$ $2 x f\left(x^2\right)$
已知 $f(x)=\left\{\begin{array}{ll}x^2 & 0 \leq x < 1 \\ 1 & 1 \leq x \leq 2\end{array}\right.$ ，设
$$
F(x)=\int_1^x f(t) \mathrm{d} t(0 \leq x \leq 2) ，
$$
则 $f(x)$ 为
$\text{A.}$ $\left\{\begin{array}{l}\frac{1}{3} x^3, 0 \leq x < 1 \\ x, 1 \leq x \leq 2\end{array}\right.$
$\text{B.}$ $\left\{\begin{array}{l}\frac{1}{3} x^3-\frac{1}{3}, 0 \leq x < 1 \\ x, 1 \leq x \leq 2\end{array}\right.$
$\text{C.}$ $\left\{\begin{array}{l}\frac{1}{3} x^3, 0 \leq x < 1 \\ x-1,1 \leq x \leq 2\end{array}\right.$
$\text{D.}$ $\begin{cases}\frac{1}{3} x^3-\frac{1}{3} & 0 \leq x < 1 \\ x-1 & 1 \leq x \leq 2\end{cases}$