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gongjy 8be42693f6 MiniMind first open source
2024-08-28 16:41:44 +08:00

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20求不定积分：$\int{\frac{x^2+1}{x^4+1}}\mathrm{d}x=$____$\dfrac{\sqrt{3}}{2}\arctan\left(\dfrac{x-\frac{1}{x}}{\sqrt{2}}\right)+C$$\dfrac{\sqrt{3}}{3}\arctan\left(\dfrac{x-\frac{1}{x}}{\sqrt{3}}\right)+C$$\dfrac{\sqrt{2}}{2}\arctan\left(\dfrac{x+\frac{1}{x}}{\sqrt{3}}\right)+C$$\dfrac{\sqrt{2}}{2}\arctan\left(\dfrac{x-\frac{1}{x}}{\sqrt{2}}\right)+C$
31求极限：$\lim_{n\rightarrow\infty}\int_0^2{\frac{x^n\ln x}{1+x^n}}\mathrm{d}x$=____$\ln 2-1$$2\ln 2-1$$\ln 2-2$$\ln 2$
42求极限：$\lim_{x\rightarrow\infty}\left(\cos\frac{1}{x}+\sin\frac{1}{x^2}\right)^{x^2}=$____$\mathrm e^{\frac12}$$\mathrm e^{\frac23}$$\mathrm e^{\frac13}$$\mathrm e^{\frac34}$
53求球体$x^2+y^2+z^2\leq R^2$与$x^2+y^2+z^2\leqslant2Rz$所围公共部分的体积____$\frac{5}{12}\pi R^3$$\frac{1}{2}\pi R^3$$\frac{7}{12}\pi R^3$$\frac{2}{3}\pi R^3$
64求极限：$\lim_{x\rightarrow\infty}\left(\frac{x^3}{x^2+1}-\frac{x^2}{x-1}\right)=$____10-1$\frac{1}{2}$
75计算定积分：$I=\int_0^1{\ln x\ln\left(1-x\right)}\mathrm{d}x=$____$2-\frac{\pi^{2}}{6}$$2-\frac{\pi^{2}}{3}$$1-\frac{\pi^{2}}{3}$$1-\frac{\pi^{2}}{6}$
86当$p$为何值时，广义积分$\int_0^{\frac{\pi}{2}}{\left|\ln\sin x\right|^p\mathrm{d}x}$是收敛的____$p>-\frac{1}{2}$时$p<-\frac{1}{2}$时$p>-\frac{2}{3}$时$p<-\frac{2}{3}$时
97二重极限$\lim_{\left(x,y\right)\to\left(0,0\right)}\frac{3\sin\left(xy\right)}{\sqrt{1+6xy}-1}=$____$\dfrac{1}{4}$1$\dfrac{3}{4}$$\dfrac{1}{2}$
108求不定积分：$I=\int{\frac{\mathrm{d}x}{\left(x^2+x+1\right)^2}}$____$\frac{5\sqrt{3}}{9}\mathrm{arc}\tan \frac{2}{\sqrt{3}}\left( x+\frac{1}{2} \right) +\frac{3x+1}{3\left( x^2+x+1 \right)}+C$$\frac{4\sqrt{3}}{9}\mathrm{arc}\tan \frac{2}{\sqrt{3}}\left( x+\frac{1}{2} \right) +\frac{3x+1}{3\left( x^2+x+1 \right)}+C$$\frac{4\sqrt{3}}{7}\mathrm{arc}\tan \frac{2}{\sqrt{3}}\left( x+\frac{1}{3} \right) +\frac{2x+3}{3\left( x^2+x+1 \right)}+C$$\frac{4\sqrt{3}}{9}\mathrm{arc}\tan \frac{2}{\sqrt{3}}\left( x+\frac{1}{2} \right) +\frac{2x+1}{3\left( x^2+x+1 \right)}+C$
119设区域$D$是由$y=x,x^2+y^2=2x$以及$x$轴所围成的第一象限的部分；求区域$D$绕$x=2$旋转所得的旋转体体积____$\frac{\pi^{2}}{4}+\frac{\pi}{3}$$\frac{\pi^{2}}{2}+\frac{\pi}{3}$$\frac{\pi^{2}}{2}+\frac{2\pi}{3}$$\frac{\pi^{2}}{2}+\frac{4\pi}{3}$
1210设区域$D_1$为$x^2+y^2\leq1$在第一象限的部分，$D_2$为$\left|x\right|+\left|y\right|\leq1$在第一象限的部分，则$I_1=\iint_{D_1}{\cos\left(\pi x y\right)}\mathrm{d}\sigma,I_2=\iint_{D_2}{\cos\left(\pi x y\right)}\mathrm{d}\sigma,I_3=\iint_{D_2}{\sin\left(\pi x y\right)}\mathrm{d}\sigma$的大小关系是？____$I_3>I_2>I_1$$I_2>I_1>I_3$$I_3>I_1>I_2$$I_1>I_2>I_3$
1311求不定积分$I=\int{\frac{1+\ln x}{x^{-x}+x^x}}\mathrm{d}x$____$\mathrm{arc}\tan \left( x^x \right) + x +C$$\mathrm{arc}\tan \left( x^x \right) +2x +C$$\mathrm{arc}\tan \left( x^x \right) - x +C$$\mathrm{arc}\tan \left( x^x \right) +C$
1412求极限：$\lim_{x\rightarrow0^+}\frac{\mathrm{e}^x-1-x}{\sqrt{1-x}-\cos\sqrt{x}}$=____0-1-2-3
1513设$f\left(x\right)$在$\left[-\frac{\pi}{2},+\infty\right)$内可导，$f\left(0\right)=a$，$f\prime\left(x\right)=\begin{cases} \frac{a}{\left(x+1\right)\left(x^2+x+1\right)}\text{，}x\geq0\\ \frac{1}{2+\cos^2x}\text{，}-\frac{\pi}{2}\leq x<0\\ \end{cases}$；试求a的值____$\frac{1}{2}$$\frac{1}{3}$$\frac{1}{4}$$\frac{1}{5}$
1614求不定积分：$\int{\frac{x^2-1}{x^4+1}}\mathrm{d}x$____$\frac{1}{\sqrt{2}}\ln \left| \frac{x^2-\sqrt{2}x+1}{x^2+\sqrt{2}x+1} \right|+C$$\frac{1}{4\sqrt{2}}\ln \left| \frac{x^2-\sqrt{2}x+1}{x^2+\sqrt{2}x+1} \right|+C$$\frac{1}{2\sqrt{2}}\ln \left| \frac{x^2-\sqrt{2}x+1}{x^2+\sqrt{2}x+1} \right|+C$$\frac{1}{3\sqrt{2}}\ln \left| \frac{x^2-\sqrt{2}x+1}{x^2+\sqrt{2}x+1} \right|+C$
1715圆柱面$x^2+y^2=1$被平面$z=0$及曲面$z=\frac{2+x+y}{1+x^2+y^2+2\left|x\right|}$截下部分的面积为？____48-412
1816求极限：$\lim_{x\rightarrow+\infty}\frac{\int_0^x{\left|\sin t\right|}\mathrm{d}t+\left|\sin x\right|\mathrm{arc}\tan x}{x}=$____$\frac{4}{\pi}$$\frac{3}{\pi}$$\frac{2}{\pi}$$\frac{1}{\pi}$
1917求极限：$\lim_{x\rightarrow0}\frac{\sin\sin\cos x-\sin\sin1}{\cos\cos\cos x-\cos\cos1}$=____$\frac{\sin\cos1\cdot\cos1}{\sin\cos1\cdot\sin1}$$\frac{\cos\sin2\cdot\cos1}{\sin\cos2\cdot\sin1}$$\frac{\cos\sin1\cdot\cos1}{\sin\cos1\cdot\sin1}$$\frac{\cos\sin1\cdot\cos2}{\sin\cos1\cdot\sin2}$
2018向量场$\boldsymbol{F}=(6x+3y+5z,12x+6y+10z,3x+3y-3z)$的散度$\mathrm{div}\boldsymbol{F}=$____$\left(6,6,-3\right)$915$\left(-7,2,9\right)$
2119已知$a_0=1,a_1=\frac{5}{4}$，$a_n=\frac{\left(2n+3\right)a_{n-1}+\left(2n-3\right)a_{n-2}}{4n}$，求极限：$\lim_{n\rightarrow\infty}a_n=$____\sqrt{\frac{1}{3}}\sqrt{\frac{2}{3}}\sqrt{\frac{3}{2}}\sqrt{\frac{3}{4}}
2220下列结论中，正确结论的个数是____。 (1)若级数$\sum_{n=1}^{\infty}a_{n}$条件收敛.则幂级数$\sum_{n=1}^{\infty}na_{n}x^{n}$的收敛半径为1； (2)若数项级数$\sum_{n=1}^{\infty}a_{n}$和$\sum_{n=1}^{\infty}b_{n}$都发散，则级数$\sum_{n=1}^{\infty}\big(|a_{n}|+|b_{n}|\big)$发散； (3)设$\left\{u_{n}\right\}$为单调递增有界数列，则$\sum_{n=1}^{\infty}\big(u_{n+1}^{2}-u_{n}^{2}\big)$收敛； (4)若$\sum_{n=1}^{\infty}nu_{n}$绝对收敛，$\sum_{n=1}^{\infty}\frac{v_{n}}{n}$条件收敛，则$\sum_{n=1}^{\infty}\left(u_{n}+v_{n}\right)$条件收敛。一个两个三个四个
2321求极限：$\lim_{x\rightarrow0}\frac{1}{x}\ln\frac{\mathrm{e}^x+\mathrm{e}^{2x}+\cdots+\mathrm{e}^{nx}}{n}=$____$\frac{n-1}{2}$$\frac{n}{2}$$\frac{n+1}{2}$$\frac{2n-1}{2}$
2422求极限：$lim_{x\rightarrow0}\left[\frac{\int_0^x{\mathrm{e}^{-t}}\cos t\mathrm{d}t}{\ln^2\left(1+x\right)}-\frac{1}{x}\right]=$____$\frac{1}{5}$$\frac{1}{4}$$\frac{1}{3}$$\frac{1}{2}$
2523计算广义积分：$\int_{-\infty}^{+\infty}{\frac{\mathrm{d}y}{\left(x^2+y^2\right)^{\frac{3}{2}}}}=$____$\frac{1}{x^2}$$\frac{2}{x^2}$$\frac{3}{x^2}$$\frac{5}{x^2}$
2624求不定积分：$I=\int{\frac{\cos2x-3}{\cos^4x\sqrt{4-\cot^2x}}\mathrm{d}x}$____$-\left[ \frac{\left( 3-\cot ^2x \right) ^{\frac{3}{2}}}{4\cot ^3x}+\frac{\sqrt{4-\cot ^2x}}{3\cot x} \right] +C$$-\left[ \frac{\left( 3-\cot ^2x \right) ^{\frac{1}{2}}}{3\cot ^3x}+\frac{\sqrt{4-\cot ^2x}}{4\cot x} \right] +C$$-\left[ \frac{\left( 4-\cot ^2x \right) ^{\frac{1}{2}}}{3\cot ^3x}+\frac{\sqrt{4-\cot ^2x}}{3\cot x} \right] +C$$-\left[ \frac{\left( 4-\cot ^2x \right) ^{\frac{3}{2}}}{3\cot ^3x}+\frac{\sqrt{4-\cot ^2x}}{4\cot x} \right] +C$
2725如果$f\prime\left(x\right)=-9x\mathrm{e}^{x^2}$并且$f\left(0\right)=-6$，则$f\left(x\right)$等于？____$-\frac{7}{2}\mathrm{e}^{x^{2}}+\frac{21}{2}$$-\frac{9}{2}\mathrm{e}^{x^{2}}+\frac{19}{2}$$-\frac{9}{2}\mathrm{e}^{x^{2}}+\frac{21}{2}$$-\frac{9}{4}\mathrm{e}^{x^{2}}+\frac{19}{2}$
2826$\lim_{x\rightarrow0}x\left[\frac{2}{x}\right]=$____0123
2927求不定积分：$\int{\frac{\mathrm{e}^x+1}{\sqrt{\mathrm{e}^x-1}}\mathrm{d}x}=$____$3\sqrt{\mathrm{e}^x-1}+2\mathrm{arctan}\sqrt{\mathrm{e}^x-1}+C$$2\sqrt{\mathrm{e}^x-1}+2\mathrm{arctan}\sqrt{\mathrm{e}^x-1}+C$$2\sqrt{\mathrm{e}^x+1}+2\mathrm{arctan}\sqrt{\mathrm{e}^x-1}+C$$2\sqrt{\mathrm{e}^x-1}+3\mathrm{arctan}\sqrt{\mathrm{e}^x+1}+C$
3028$\lim_{n\rightarrow\infty}\frac{3^n+2^n}{3^{n+1}-2^{n+1}}=$____$\frac{2}{3}$$\frac{1}{3}$$\frac{3}{4}$$\frac{2}{5}$
3129计算广义积分：$I=\int_1^{+\infty}{\frac{1}{\sqrt{x}}\ln\left(\frac{x+1}{x}\right)}\mathrm{d}x$____$\pi -\ln 2$$\pi -2\ln 2$$\pi -3\ln 2$$2\pi -\ln 2$
3230计算不定积分：$I=\int{\frac{\mathrm{d}x}{x^2+2x+3}}$____$\frac{\sqrt{2}}{4}\mathrm{arc}\tan \left( \frac{x+1}{\sqrt{2}} \right) +C$$\frac{\sqrt{2}}{3}\mathrm{arc}\tan \left( \frac{x+1}{\sqrt{2}} \right) +C$$\frac{\sqrt{2}}{2}\mathrm{arc}\tan \left( \frac{x+1}{\sqrt{2}} \right) +C$$\frac{\sqrt{2}}{2}\mathrm{arc}\tan \left( \frac{x+1}{2\sqrt{2}} \right) +C$
3331求不定积分：$\int{\frac{x^2}{x^2-1-\sqrt{1-x^2}}}\mathrm{d}x$____$x-\mathrm{arc}\sin x+C$$x-\mathrm{arc}\cos x+C$$x+\mathrm{arc}\sin x+C$$x+\mathrm{arc}\cos x+C$
3432设$x=\frac{1}{u}+\frac{1}{v},y=\frac{1}{u^2}+\frac{1}{v^2},z=\frac{1}{u^3}+\frac{1}{v^3}+\mathrm{e}^x$,求$\frac{\partial z}{\partial y},\frac{\partial z}{\partial v}$____$\frac{3}{4u}+\frac{u}{2}\mathrm{e}^x,\frac{\partial z}{\partial v}=\frac{3}{uv^3}-\frac{3}{v^4}+\frac{u-v}{v^3}\mathrm{e}^x$$\frac{3}{2u}+\frac{u}{4}\mathrm{e}^x,\frac{\partial z}{\partial v}=\frac{3}{uv^3}-\frac{3}{v^4}+\frac{u-v}{v^3}\mathrm{e}^x$$\frac{3}{2u}+\frac{u}{2}\mathrm{e}^x,\frac{\partial z}{\partial v}=\frac{3}{uv^3}-\frac{3}{v^4}+\frac{u-v}{v^3}\mathrm{e}^x$$\frac{3}{4u}+\frac{u}{2}\mathrm{e}^x,\frac{\partial z}{\partial v}=\frac{3}{uv^3}-\frac{3}{v^4}+\frac{2u-v}{v^3}\mathrm{e}^x$
3533求不定积分：$\int{\frac{1}{\mathrm{e}^x-1}}\mathrm{d}x$____$\ln \left| \mathrm{e}^x-1 \right|-x+C$$\ln \left| \mathrm{e}^x-2 \right|-x+C$$\ln \left| \mathrm{e}^x-3 \right|-x+C$$\ln \left| \mathrm{e}^x-1 \right|-2x+C$
3634求极限：$\lim_{n\rightarrow\infty}\sum_{k=1}^{2n}{\frac{\sqrt{1+\sin\frac{\pi k}{n}}}{n+\frac{1}{k}}}=$____$\frac{3\sqrt{2}}{\pi}$$\frac{4\sqrt{2}}{\pi}$$\frac{5\sqrt{2}}{\pi}$$\frac{6\sqrt{2}}{\pi}$
3735利用泰勒展开求解极限：$\lim_{x\rightarrow0}\frac{\sinh x-\tanh x}{x^3}=$____$\frac{1}{2}$$\frac{1}{3}$$\frac{1}{4}$$\frac{1}{5}$
3836求极限：$\lim_{x\rightarrow1}\frac{\left(1+\frac{1}{x}\right)^x\left(1+x\right)^{\frac{1}{x}}-4}{\left(x-1\right)^2}$=____$2\ln 2-1$$3\ln 2-2$$4\ln 2-3$$5\ln 2-3$
3937求极限：$\lim_{n\rightarrow\infty}\sqrt[n^2]{2}\cdot\sqrt[n^2]{2^2}\cdot\cdots\cdot\sqrt[n^2]{2^n}=$____1$\sqrt{2}$$\sqrt{3}$$\frac{\sqrt{3}}{2}$
4038已知曲面$\Sigma$是由曲线段$\left\{\begin{array}{ll}z=\sqrt{y-1}&(1\leq y\leq4)\\x=0&\end{array}\right.$绕y轴旋转而成的旋转曲面，去左侧，则曲面积分$\iint_{\Sigma}\frac{x\mathrm{d}y\mathrm{d}z+3y\mathrm{d}z\mathrm{d}x+11z\mathrm{d}x\mathrm{d}y}{1+\sqrt{y-x^{2}-z^{2}}}=$____$\frac{405\pi}{4}$$\frac{405\pi}{2}$$\frac{63\pi}{4}$$\frac{189\pi}{4}$
4139向量场${\boldsymbol{F}}=(x^{2}+y^{2},y^{2}+z^{2},z^{2}+x^{2})$在点$\left(5,5,5\right)$处的旋度$\mathbf{rot}\boldsymbol{F}\big|_{(5,\:5,\:5)}=$____10-30$\left(-\:10\:,\:-\:10\:,\:-\:10\right)$$(-\:10\:,10\:,\:-\:10)$
4240求极限：$L=\lim_{n\rightarrow\infty}\left(\int_1^2{\sqrt[n]{1+x}}\mathrm{d}x\right)^n$=____$\frac{27}{4\mathrm{e}}$$\frac{27}{2\mathrm{e}}$$\frac{27}{5\mathrm{e}}$$\frac{9}{2\mathrm{e}}$
4341计算三重积分：$I=\iiint_V{\left(x^2+y^2\right)}\mathrm{d}x\mathrm{d}y\mathrm{d}z$；其中$V$是由曲面$2\left(x^2+y^2\right)=z$与$z=4$为界面的区域____$\frac{4\pi}{3}$$\frac{8\pi}{3}$$3\pi$$\frac{16\pi}{3}$
4442求极限：$\lim_{x\rightarrow0}\frac{\sqrt{1+x}-\sqrt[3]{1+2x^2}}{\ln\left(1+3x\right)}=$____$\frac{1}{2}$$\frac{1}{3}$$\frac{1}{4}$$\frac{1}{6}$
4543求极限：$L=\lim_{x\rightarrow0}\frac{\ln^2\left(x+\sqrt{1+x^2}\right)-1+\mathrm{e}^{-x^2}}{x^4}=$____$\frac{1}{2}$$\frac{1}{3}$$\frac{1}{5}$$\frac{1}{6}$
4644求极限：$\lim_{n\rightarrow\infty}\sum_{k=1}^n{\frac{n+k}{n^2+k}}=$____$\frac{1}{2}$1$\frac{3}{2}$2
4745若四次齐次函数$f\left(x,y,z\right)$满足$f_{xx}+f_{yy}+f_{zz}=x^2+y^2+z^2$；计算:$I=\oiint_{\Sigma}{f\left(x,y,z\right)}\mathrm{d}S=$，其中$\Sigma:x^2+y^2+z^2=1$____$\frac{\pi}{5}$$\frac{2\pi}{5}$$\frac{3\pi}{5}$$\frac{4\pi}{5}$
4846求不定积分：$I=\int{\frac{x^2}{\left(x\cos x-\sin x\right)\left(x\sin x+\cos x\right)}\mathrm{d}x}$____$\ln \left| \frac{x\cos x+\sin x}{x\cos x-\sin x} \right|+C$$\ln \left| \frac{x\sin x+\cos x}{x\cos x-\sin x} \right|+C$$\ln \left| \frac{x\cos x+\sin x}{x\cos x-\cot x} \right|+C$$\ln \left| \frac{x\cos x+\sin x}{x\sin x-\cos x} \right|+C$
4947计算二重积分：$\iint_D{\frac{1}{\sqrt{x^2+y^2}}\mathrm{d}x\mathrm{d}y}=$，其中$D=\left\{\left(x,y\right)|0\leq x\leq y\leq1\right\}$____$\ln\left(1+2\sqrt{2}\right)$$\ln\left(2+\sqrt{2}\right)$$\ln\left(1+\sqrt{2}\right)$$\ln\left(2+2\sqrt{2}\right)$
5048设曲线$y=x\sin x$在$\left[0,n\pi\right]\left(n=1,2,\cdots\right)$上与$x$轴所围成的面积为$S_n$，求极限：$\lim_{n\rightarrow\infty}\left[\frac{S_n}{\pi\left(n+1\right)^2}\right]^n$____$\mathrm{e}^{-1}$$\mathrm{e}^{-2}$$\mathrm{e}^{-3}$$\mathrm{e}^{-\frac{3}{2}}$
5149将函数$f\left(x\right)=\frac{1}{4}\ln\frac{1+x}{1-x}+\frac{1}{2}\mathrm{arc}\tan x-x$展开成幂级数____$\begin{aligned}\sum_{k=1}^n\frac{x^{3k+1}}{3k+1},&|x|<1\end{aligned}$$\begin{aligned}\sum_{k=1}^n\frac{2x^{3k+1}}{3k+1},&|x|<1\end{aligned}$$\begin{aligned}\sum_{k=1}^n\frac{x^{4k+1}}{4k+1},&|x|<1\end{aligned}$$\begin{aligned}\sum_{k=1}^n\frac{2x^{4k+1}}{4k+1},&|x|<1\end{aligned}$
5250$z=\mathrm{e}^{-4x}\mathrm{ln}\left(6y\right)$在点$\left(0,1\right)$处的全微分$\mathrm{d}z\big|_{(0,1)}=$____$-4\ln6\mathrm{d}x+\mathrm{d}y$$4\ln6\mathrm{d}x-\mathrm{d}y$$-4\ln6\mathrm{d}x-\mathrm{d}y$$-4\ln6\mathrm{d}x+6\mathrm{d}y$
5351求极限：$\lim_{n\rightarrow\infty}\left(2n+1\right)\left\{\frac{\pi}{2}-\frac{1}{2n+1}\left[\frac{\left(2n\right)!!}{\left(2n-1\right)!!}\right]^2\right\}$=____$\frac{\pi}{4}$$\frac{\pi}{2}$$\frac{\pi}{3}$$\frac{2\pi}{3}$
5452求定积分：$\int_0^2{\left(x-1\right)^2\sqrt{2x-x^2}}\mathrm{d}x=$____$\frac{\pi}{4}$$\frac{\pi}{8}$$\frac{\pi}{12}$$\frac{\pi}{3}$
5553求极限：$\lim_{n\rightarrow\infty}\frac{\sqrt[2]{n}+\sqrt[3]{n}+\cdots+\sqrt[n]{n}}{n}$=____01-1$\frac{1}{2}$
5654已知：$f\left(x\right)=\sin x+\int_0^{\frac{\pi}{4}}{f\left(2x\right)}\mathrm{d}x$，求$f\left(x\right)$____$\sin x+\frac{1}{1-\frac{\pi}{2}}$$\cos x+\frac{1}{2-\frac{\pi}{2}}$$\sin x+\frac{1}{3-\frac{\pi}{2}}$$\sin x+\frac{1}{2-\frac{\pi}{2}}$
5755求极限：$\lim_{x\rightarrow0}\frac{\left(1+x\right)^{\frac{1}{x}}-\left(1+2x\right)^{\frac{1}{2x}}}{\sin x}$=____$\frac{\mathrm{e}}{2}$$\frac{\mathrm{e}}{3}$$\frac{\mathrm{e}}{4}$$\frac{3\mathrm{e}}{4}$
5856设数列$\left\{a_n\right\}$单调减少，$\lim_{n\rightarrow\infty}a_n=0,s_n=\sum_{k=1}^n{a_k}\left(n=1,2,3,\cdots\right)$无界，则幂级数$\sum_{n=1}^{\infty}{a_n\left(x-1\right)^n}$的收敛半径为?____$\left( -1,1 \right]$$\left[ -1,1 \right)$$\left[ 0,2 \right)$$\left( 0,2 \right]$
5957求不定积分：$I=\int{\frac{\mathrm{e}^x}{\mathrm{e}^{2x}\left(1+\mathrm{e}^{2x}\right)}}\mathrm{d}x$____$-\frac{4}{\mathrm{e}^x}-\mathrm{arc}\tan\mathrm{e}^x+C$$-\frac{3}{\mathrm{e}^x}-\mathrm{arc}\tan\mathrm{e}^x+C$$-\frac{2}{\mathrm{e}^x}-\mathrm{arc}\tan\mathrm{e}^x+C$$-\frac{1}{\mathrm{e}^x}-\mathrm{arc}\tan\mathrm{e}^x+C$
6058函数$f(x)={\frac{\sqrt{6}x}{\left(x-1\right)^{2}}}+3\ln\left(1-x\right)$在$x=0$处的幂级数展开式为____$f(x)=\sum_{n=1}^{\infty}\left({\sqrt{6}}n-{\frac{3}{n}}\right)x^{n},-1<x<1$$f(x)=\sum_{n=1}^{\infty}\Bigl(\sqrt{6}n+\frac{3}{n}\Bigr)x^{n},-1<x<1$$f(x)=\sum_{n=1}^{\infty}\Bigl(\sqrt{6}n+\dfrac{3}{n}\Bigr)x^{n-1},-1<x<1$$f(x)=\sum_{n=1}^{\infty}\Bigl(\sqrt{6}n-\frac{3}{n}\Bigr)x^{n-1},-1<x<1$
6159计算$\iiint_{\Omega}{\left(y^2+z^2\right)}\mathrm{d}x\mathrm{d}y\mathrm{d}z=$其中$\Omega$是由$xOy$平面上的曲线$y^2=2x$绕$x$轴旋转而成的曲面与$x=5$所围成的区域____$\frac{200\pi}{3}$$\frac{250\pi}{3}$$100\pi$$\frac{320\pi}{3}$
6260$x_n=\sum_{k=1}^n{\left[\left(n^k+1\right)^{-\frac{1}{k}}+\left(n^k-1\right)^{\frac{1}{k}}\right]$，求$\lim_{n\rightarrow\infty}x_n}$=____0123
6361求不定积分：$I=\int{\sqrt{1+\frac{1}{x^2}}}\mathrm{d}x$____$\sqrt{1+x^2}+\ln \left| \frac{\sqrt{4+x^2}-1}{x} \right|+C$$\sqrt{1+x^2}+\ln \left| \frac{\sqrt{1+2x^2}-1}{x} \right|+C$$\sqrt{1+x^2}+\ln \left| \frac{\sqrt{1+x^2}-1}{x} \right|+C$$\sqrt{1+x^2}+\ln \left| \frac{\sqrt{1+2x^2}-1}{x} \right|+C$
6462求极限：$\lim_{x\rightarrow+\infty}x^2\left(\mathrm{arc}\tan\frac{1}{x}-\mathrm{arc}\tan\frac{1}{x+1}\right)=$____-101$\frac{2}{3}$
6563求不定积分：$I=\int{\sqrt{1+x^2}}\mathrm{d}x$____$\frac{1}{5}\left( x\sqrt{1+x^2}+\ln \left| x+\sqrt{1+x^2} \right| \right) +C$$\frac{1}{4}\left( x\sqrt{1+x^2}+\ln \left| x+\sqrt{1+x^2} \right| \right) +C$$\frac{1}{3}\left( x\sqrt{1+x^2}+\ln \left| x+\sqrt{1+x^2} \right| \right) +C$$\frac{1}{2}\left( x\sqrt{1+x^2}+\ln \left| x+\sqrt{1+x^2} \right| \right) +C$
6664求极限：$\lim_{x\rightarrow0}\frac{\tan\tan\tan x-\sin\sin\sin x}{\tan x-\sin x}=$____3421
6765求不定积分：$I=\int{\left(x-2\right)\sqrt{x^2+4x+1}}\mathrm{d}x$____$\frac{\left( x^2+4x+1 \right) ^{\frac{3}{2}}}{3}-2\left( x+2 \right) \sqrt{x^2+4x+1}-6\ln \left| \frac{x+2+\sqrt{x^2+4x+1}}{\sqrt{3}} \right|+C$$\frac{\left( x^2+4x+1 \right) ^{\frac{3}{2}}}{4}-2\left( x+2 \right) \sqrt{x^2+4x+1}-3\ln \left| \frac{x+2+\sqrt{x^2+4x+1}}{\sqrt{3}} \right|+C$$\frac{\left( x^2+4x+1 \right) ^{\frac{3}{2}}}{4}-2\left( x+2 \right) \sqrt{x^2+4x+1}-4\ln \left| \frac{x+2+\sqrt{x^2+4x+1}}{\sqrt{3}} \right|+C$$\frac{\left( x^2+3x+1 \right) ^{\frac{3}{2}}}{4}-2\left( x+2 \right) \sqrt{x^2+4x+1}-3\ln \left| \frac{x+2+\sqrt{x^2+4x+1}}{\sqrt{3}} \right|+C$
6866求不定积分：$I=\int{\cot^2x}\tan\left(\frac{1+x\tan x}{\tan x}\right)\mathrm{d}x$____$\ln \left[ \cos \left( x+\sin x \right) \right] +C$$\ln \left[ \cos \left( x+\cos x \right) \right] +C$$\ln \left[ \cos \left( x+\tan x \right) \right] +C$$\ln \left[ \cos \left( x+\cot x \right) \right] +C$
6967计算定积分：$\int_1^3{\sqrt{1+\frac{1}{4}\left(x-\frac{1}{x}\right)^2}}\mathrm{d}x$____$3+\frac{1}{2}\ln 3$$2+\frac{1}{3}\ln 3$$3+\frac{1}{3}\ln 3$$2+\frac{1}{2}\ln 3$
7068计算定积分：$I=\int_0^{\pi}{\sqrt{\sin x-\sin^2x}\mathrm{d}x}$____$\sqrt{2}+\ln \left( \frac{\sqrt{2}-1}{\sqrt{2}+1} \right)$$\sqrt{2}-\ln \left( \frac{\sqrt{2}-1}{\sqrt{2}+1} \right)$$2\sqrt{2}+\ln \left( \frac{\sqrt{2}-1}{\sqrt{2}+1} \right)$$2\sqrt{2}-\ln \left( \frac{\sqrt{2}-1}{\sqrt{2}+1} \right)$
7169设$f\left(x\right)$在$\left[0,1\right]$上连续可导，且$f\left(0\right)=f\left(1\right)=0$则存在常数$c>0$使得：$\int_0^1{f^2\left(x\right)}\mathrm{d}x\leq c\int_0^1{\left[f\prime\left(x\right)\right]^2}\mathrm{d}x$，则$c$最小为？____$\frac{1}{\pi ^2}$$\frac{1}{2\pi ^2}$$\frac{2}{\pi ^2}$$\frac{1}{\pi}$
7270设可微函数$f\left(x,y\right)$在点$P\left(x,y\right)$处沿$\boldsymbol{l}_1\left(1,1\right)$的方向导数为$\frac{\sqrt{2}}{2}\cdot\frac{2xy-x^2}{x^4+y^2}$，沿$\boldsymbol{l}_2\left(0,-2\right)$的方向导数为$\frac{x^2}{x^4+y^2}$,若$f\left(1,1\right)=0$，求函数$f\left(x,y\right)$____$f\left( x,y \right) =-\mathrm{arc}\tan \left( \frac{y}{2x^2} \right)$$f\left( x,y \right) =-\mathrm{arc}\tan \left( \frac{y}{x^2} \right)$$f\left( x,y \right) =-\mathrm{arc}\tan \left( \frac{3y}{2x^2} \right)$$f\left( x,y \right) =-\mathrm{arc}\tan \left( \frac{2y}{x^2} \right)$
7371计算曲线积分：$\oint_C{\left(y-z\right)\mathrm{d}x+\left(z-x\right)\mathrm{d}y+\left(x-y\right)\mathrm{d}z}=$。其中$C$为$x^2+y^2=1$与$x+y+z=1$的交线，从$x$轴正向看是逆时针方向____$-3\pi$$-6\pi$$-9\pi$$-12\pi$
7472设$f\left(x\right)=\int_1^x{\frac{\ln t}{1+t}}\mathrm{d}t\left(x>0\right)$，求$f\left(2\right)+f\left(\frac{1}{2}\right)$的值____$\frac{1}{6}\ln ^22$$\frac{1}{4}\ln ^22$$\frac{1}{2}\ln ^22$$\frac{1}{3}\ln ^22$
7573计算第二型曲线积分$\int_S{\omega}=$其中：曲线$S=\left\{\left(x,y\right)|y=\sin x,x\in\left[0,\pi\right]\right\}$，定向沿着$x$增大的方向，$\omega=\mathrm{e}^x\left(1-2\cos y\right)\mathrm{d}x-2\mathrm{e}^x\left(y-\sin y\right)\mathrm{d}y=$____$-\frac{4}{5}\left( \mathrm{e}^{\pi}-1 \right)$$-\frac{3}{5}\left( \mathrm{e}^{\pi}-1 \right)$$-\frac{3}{4}\left( \mathrm{e}^{\pi}-1 \right)$$-\frac{3}{7}\left( \mathrm{e}^{\pi}-1 \right)$
7674求极限：$\lim_{n\rightarrow\infty}\frac{\sum_{i=1}^n{i^k}}{n^{k+1}}=$____，其中k>0。$\dfrac{1}{k+1}$$\dfrac{1}{k}$$\dfrac{1}{k-1}$$\dfrac{2}{k+1}$
7775求不定积分：$\int{-10\cot x\ln\left(\sin x\right)}\mathrm{d}x=$____$-2\ln^2\left(\sin x\right)+C$$-3\ln^2\left(\sin x\right)+C$$-4\ln^2\left(\sin x\right)+C$$-5\ln^2\left(\sin x\right)+C$
7876设$\int_0^1{\frac{x^{n-1}}{1+x}}\mathrm{d}x=\frac{a}{n}+\frac{b}{n^2}+o\left(\frac{1}{n^2}\right),n\rightarrow\infty$，求$a,b$____$a=\frac{1}{3}$,$b=\frac{1}{4}$$a=\frac{1}{2}$,$b=\frac{1}{3}$$a=\frac{1}{2}$,$b=\frac{1}{4}$$a=\frac{1}{4}$,$b=\frac{1}{2}$
7977二重极限$\operatorname*{lim}_{(x,y)\to(0,\:1)}\biggl(1+{\frac{x}{3y}}\biggr)^{\frac{1}{1+5xy}}=$____$\mathrm{e}^{1/36}$$\mathrm{e}^{1/18}$$\mathrm{e}^{1/48}$$\mathrm{e}^{1/9}$
8078$\lim_{x\rightarrow a}\frac{x^a-a^x}{x^2-a^2}\left(a>0\right)=$____$\frac{a^{a+1}(1-\operatorname{ln}a)}{2}$$\frac{a^{a-1}(1-\operatorname{ln}a)}{2}$$\frac{a^{a-1}(1-\operatorname{ln}a)}{3}$$\frac{(a-1)^a(1-\operatorname{ln}a)}{2}$
8179设$D$：$\left\{\left(x,y\right)|x^2+y^2\leq4\right\}$，计算$I=\iint_D{\left|2x-x^2-y^2\right|}\mathrm{d}x\mathrm{d}y=$____$7\pi$$8\pi$$9\pi$$10\pi$
8280求不定积分：$I=\int{\frac{\cos x}{2\sin x+3\cos x}\mathrm{d}x}$____$\frac{3}{13}x+\ln \left| 2\sin x+3\cos x \right|+C$$\frac{1}{4}x+\ln \left| 2\sin x+3\cos x \right|+C$$\frac{5}{13}x+\ln \left| 2\sin x+3\cos x \right|+C$$\frac{6}{13}x+\ln \left| 3\sin x+2\cos x \right|+C$
8381设幂级数$\sum_{n=1}^{\infty}na_{n}\left(x-6\right)^{n}$的收敛区间为$(-\:2\:,14)$，则幂级数$\sum_{n=1}^{\infty}a_{n}\left(x+5\right)^{2n}$的收敛区间为____$\left(-5-2\sqrt{2}\:,5+2\sqrt{2}\right)$$\left(5-2{\sqrt{2}}\:,5+2{\sqrt{2}}\right)$$\left(-13,3\right)$$\left(-\:5\:-\:2\sqrt{2}\:,\:-\:5+2\sqrt{2}\:\right)$
8482求不定积分：$I=\int{\frac{\tan x}{1+\tan^3x}}\mathrm{d}x$____$\frac{1}{4}\ln \left( \sec x \right) -\frac{1}{4}x-\frac{1}{6}\ln \left| \sec x-\tan x \right|+\frac{\sqrt{3}}{3}\mathrm{arc}\tan \left( \frac{2\tan x-1}{\sqrt{3}} \right) -\frac{1}{6}\ln \left| 1+\tan x \right|+C$$\frac{1}{4}\ln \left( \sec x \right) -\frac{1}{3}x-\frac{1}{3}\ln \left| \sec x-\tan x \right|+\frac{\sqrt{3}}{3}\mathrm{arc}\tan \left( \frac{2\tan x-1}{\sqrt{3}} \right) -\frac{1}{6}\ln \left| 1+\tan x \right|+C$$\frac{1}{2}\ln \left( \sec x \right) -\frac{1}{2}x-\frac{1}{6}\ln \left| \sec x-\tan x \right|+\frac{\sqrt{3}}{3}\mathrm{arc}\tan \left( \frac{2\tan x-1}{\sqrt{3}} \right) -\frac{1}{6}\ln \left| 1+\tan x \right|+C$$\frac{1}{3}\ln \left( \sec x \right) -\frac{1}{2}x-\frac{1}{4}\ln \left| \sec x-\tan x \right|+\frac{\sqrt{3}}{3}\mathrm{arc}\tan \left( \frac{2\tan x-1}{\sqrt{3}} \right) -\frac{1}{6}\ln \left| 1+\tan x \right|+C$
8583已知平面区域D由直线$\frac{x}{\sqrt{6}}+{\frac{y}{3}}=1$、$x=0$和$y=0$围成，曲片面$\Sigma\:z\:=\:\sin\left(x+y\right)$，其中$(x,y)\:\in\:D$，则曲面积$\iint_{\Sigma}{\frac{1}{\sqrt{3-2z^{2}}}}\mathrm{d}S=$____$\frac{3\sqrt{\frac{3}{2}}}{2}$$3\sqrt{6}$$\frac{9\sqrt{\frac{3}{2}}}{2}$$3\sqrt{\dfrac{3}{2}}$
8684计算定积分：$I=\int_{-\pi}^{\pi}{\frac{x\sin x\cdot\mathrm{arc}\tan e^x}{1+\cos^2x}}\mathrm{d}x$____$\dfrac{\pi^{3}}{8}$$\dfrac{\pi^{3}}{6}$$\dfrac{\pi^{3}}{4}$$\dfrac{\pi^{2}}{2}$
8785计算二重积分：$\iint_D{\ln\left(\frac{x}{y^2}\right)}\mathrm{d}x\mathrm{d}y=$，其中$D=\left\{\sqrt{x}+\sqrt{y}\leq1,x\geq0,y\geq0\right\}$____$\frac{13}{36}$$\frac{5}{12}$$\frac{4}{9}$$\frac{19}{36}$
8886当$x\rightarrow0$时，下列最高阶的无穷小量是？$\left(1\right)\ln\left(1+x^2\right)-x^2；\left(2\right)\sqrt{1+x^2}+\cos x-2；\left(3\right)\int_0^{x^2}{\ln\left(1+t^2\right)}\mathrm{d}t；\left(4\right)\mathrm{e}^{x^2}-1-x^2$____$\left( 1 \right)$$\left( 2 \right)$$\left( 3 \right)$$\left( 4 \right)$
8987求极限：$\lim_{n\rightarrow\infty}\sqrt{n}\underset{n\text{个}}{\underbrace{\sin\sin\cdots\sin x}}\text{=，}x\in\left(0,1\right)$____1$\sqrt{2}$$\sqrt{3}$$\sqrt{5}$
9088求极限：$\lim_{x\rightarrow0}\frac{\tan\left(\mathrm{e}^x-1\right)-\mathrm{e}^{\tan x}+1}{x^4}$=____$\frac{1}{3}$$\frac{1}{6}$$\frac{1}{9}$$\frac{1}{12}$
9189求二重积分$\iint\limits_D{\left(2-x-y\right)\mathrm{d}x\mathrm{d}y}=$其中$D:\left(x-1\right)^2+\left(y-1\right)^2+\left(5-x-y\right)^2\leq12$____$-6\sqrt{3}\pi$$-5\sqrt{3}\pi$$-4\sqrt{3}\pi$$-3\sqrt{3}\pi$
9290求不定积分：$\int{\frac{x^2+x+1}{x\left(1+x^2\right)\left(1+x\mathrm{e}^{\mathrm{arc}\tan x}\right)}}\mathrm{d}x$____$\ln \left| \frac{x\mathrm{e}^{\mathrm{arc}\tan x}}{2+x\mathrm{e}^{\mathrm{arc}\tan x}} \right|+C$$\ln \left| \frac{x\mathrm{e}^{\mathrm{arc}\tan x}}{1+2x\mathrm{e}^{\mathrm{arc}\tan x}} \right|+C$$\ln \left| \frac{2x\mathrm{e}^{\mathrm{arc}\tan x}}{2+x\mathrm{e}^{\mathrm{arc}\tan x}} \right|+C$$\ln \left| \frac{x\mathrm{e}^{\mathrm{arc}\tan x}}{1+x\mathrm{e}^{\mathrm{arc}\tan x}} \right|+C$
9391求极限：$\lim_{x\rightarrow a}\sin\left(\frac{x-a}{2}\right)\tan\left(\frac{\pi x}{2a}\right)=$____$-\dfrac{a}{\pi}$$-\dfrac{2a}{\pi}$$-\dfrac{a}{2\pi}$$-\dfrac{2a}{3\pi}$
9492求极限：$\lim_{n\rightarrow\infty}\frac{5n+2\sqrt{n}+4}{\sqrt{n^2+1}}=$____2345
9593计算第二型曲线积分$\int_S{\omega}=$其中：曲线$S=\left\{\left(x,y,z\right)|x^2+y^2+z^2=1,x+y+z=0\right\}$，从第一象限看定向为逆时针方向，____$-2\sqrt{3}\pi$$2\sqrt{3}\pi$$-3\sqrt{3}\pi$$3\sqrt{3}\pi$
9694求定积分：$\int_0^{\frac{\pi}{3}}{\tan x}\mathrm{d}x=$____$\text{ln}2$$\text{ln}3$$\text{ln}4$$\text{ln}5$
9795设函数$f\left(x\right)$满足：$xf\prime\left(x\right)-f\left(x\right)=\sqrt{2x-x^2}$，且$f\left(1\right)=0$，求：$\int_0^1{f\left(x\right)}\mathrm{d}x$____$-\frac{\pi}{8}$$-\frac{\pi}{4}$$-\frac{\pi}{2}$$-\frac{2\pi}{3}$
9896求不定积分：$I=\int{x\sin^2x}\mathrm{d}x$____$\frac{1}{2}-\frac{1}{4}x\sin 2x-\frac{1}{8}\cos 2x+C$$\frac{1}{4}-\frac{1}{2}x\sin 2x-\frac{1}{4}\cos 2x+C$$\frac{1}{4}-\frac{1}{4}x\sin 2x-\frac{1}{8}\cos 2x+C$$\frac{1}{2}-\frac{1}{4}x\sin 2x-\frac{1}{4}\cos 2x+C$
9997计算定积分：$I=\int_1^{\sqrt{3}}{\frac{\mathrm{d}x}{x^2\sqrt{1+x^2}}}$____$4\left( \frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}} \right)$$3\left( \frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}} \right)$$2\left( \frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}} \right)$$\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}$
10098计算二重积分：$I=\int_0^{a\sin\varphi}{\mathrm{e}^{-y^2}}\mathrm{d}y\int_{\sqrt{a^2-y^2}}^{\sqrt{b^2-y^2}}{\mathrm{e}^{-x^2}}\mathrm{d}x+\int_{a\sin\varphi}^{b\sin\varphi}{\mathrm{e}^{-y^2}}\mathrm{d}y\int_{y\cot\varphi}^{\sqrt{b^2-y^2}}{\mathrm{e}^{-x^2}}\mathrm{d}x=$其中$0<a<b,0<\varphi<\frac{\pi}{2}$____$\dfrac{\mathrm{e}^{-a^2}-\mathrm{e}^{-b^2}}{4}\varphi$$\dfrac{\mathrm{e}^{-a^2}-\mathrm{e}^{-b^2}}{3}\varphi$$\dfrac{\mathrm{e}^{-a^2}-\mathrm{e}^{-b^2}}{2}\varphi$$\dfrac{2\mathrm{e}^{-a^2}-\mathrm{e}^{-b^2}}{3}\varphi$
10199函数$u=x^2+y^2+2z^2$在点$P\left(1,1,\sqrt{2}\right)$处沿曲线$\left\{\begin{array}{l} x^2+y^2+z^2=4\\ x^2+y^2=2x\\ \end{array}\right.$在该点处指向$x$轴正向一侧切线方向的方向导数为？____$-\frac{\sqrt{6}}{12}$$-\frac{\sqrt{6}}{6}$$-\frac{\sqrt{6}}{3}$$-\frac{2\sqrt{6}}{3}$
102100计算定积分：$\int_0^1{\ln^{2022}x\mathrm{d}x}$____$\begin{array}{c}1011!\end{array}$$\begin{array}{c}2022!\end{array}$$\begin{array}{c}2023!\end{array}$$\begin{array}{c}4044!\end{array}$
103101设$f\left(x\right)=\int_1^x{\frac{\ln\left(1+t\right)}{t}}\mathrm{d}t$，则$\int_0^1{\frac{f\left(x\right)}{\sqrt{x}}}\mathrm{d}x$为？____$4-4\ln 2-2\pi$$8-4\ln 2-2\pi$$4-2\ln 2-2\pi$$4-4\ln 2-2\pi$
104102已知函数$f(x,y)$在$\mathbb{R}^{2}$上连续，且$f(x,y)=3xy+\iint_{D}f(x,y)\mathrm{d}x\mathrm{d}y$，其中平面区域D由$y=0$，$y=2x^2$以及$x=1$围成，则$f(x,y)$____3xy+93xy+123xy+153xy+3
105103设函数$f\left(x\right)$连续，且$f\left(0\right)\ne0$，求极限：$\lim_{x\rightarrow0}\frac{x\int_0^x{f\left(x-t\right)}\mathrm{d}t}{\int_0^x{tf\left(x-t\right)}\mathrm{d}t}=$____1234
106104设$G_n=\sqrt[n+1]{\prod_{k=1}^n{C_{n}^{k}}}$，其中$C_{n}^{k}=\frac{n!}{k!\left(n-k\right)!}$，求$\lim_{n\rightarrow\infty}\sqrt[n]{G_n}$____$\sqrt{\text{2e}}$$\sqrt{\text{3e}}$$\sqrt{\text{e}}$$\sqrt{\text{4e}}$
107105设$\Sigma$为上半球面$x^2+y^2+z^2=a^2\left(a>0,z\geq0\right)$，则积分$\iint_{\Sigma}{\left(\sqrt{2}x+z+1\right)^2}\mathrm{d}S$为？____$3\pi a^2\left(a^2+a+1\right)$$2\pi a^2\left(a^2+a+2\right)$$2\pi a^2\left(a^2+3a+1\right)$$2\pi a^2\left(a^2+a+1\right)$
108106计算定积分：$I=\int_{-\pi}^{\pi}{\frac{x\sin x\left(\mathrm{arc}\tan\mathrm{e}^x+\int_0^x{\mathrm{e}^{t^2}\mathrm{d}t}\right)}{1+\cos^2x}}\mathrm{d}x$____$\dfrac{\pi^{3}}{12}$$\dfrac{\pi^{3}}{8}$$\dfrac{\pi^{3}}{4}$$\dfrac{\pi^{3}}{2}$
109107设$L$：$\left\{\begin{array}{l} x^2+y^2+z^2=a^2\\ y+z=a\\ \end{array}\right.$，试计算：$\oint_L{\left(x^2+y^2+z^2\right)\mathrm{d}s}=$____$\sqrt{2}\pi a^3$$\sqrt{3}\pi a^3$$\sqrt{5}\pi a^3$$\sqrt{6}\pi a^3$
110108求极限：$\lim_{n\rightarrow\infty}\sqrt[n]{\frac{\left(2n\right)!}{n^nn!}}$=____$\frac{2}{\mathrm{e}}$$\frac{4}{\mathrm{e}}$$\frac{6}{\mathrm{e}}$$\frac{8}{\mathrm{e}}$
111109设$f\left(x\right)$连续，且$f\left(0\right)=0,f\prime\left(0\right)\ne0$，求$\lim_{t\rightarrow0^+}\frac{\iint_D{f\left(\sqrt{x^2+y^2}\right)}\mathrm{d}x\mathrm{d}y}{\int_0^t{yf\left(t-y\right)}\mathrm{d}y}$，其中$D:\left\{\left(x,y\right)|x^2+y^2\leq t^2\right\}$____$4\pi$$3\pi$$2\pi$$\pi$
112110已知$\tan^2\theta=2\tan\theta+1$,$\left(0<\theta<\pi\right)$,则，$\tan\theta$=____$2+\tan 3\theta \$$3+\tan 2\theta \$$3+\tan 3\theta \$$4+\tan 3\theta \$
113111平面${\frac{x}{2}}+{\frac{y}{2}}+{\frac{z}{\sqrt{5}}}=1$位于第一卦限部分的面积为____$\sqrt{\frac{7}{2}}$$3\sqrt{\dfrac{7}{2}}$$\frac{2\sqrt{14}}{3}$$\sqrt{14}$
114112求不定积分：$I=\int{\frac{\mathrm{d}x}{\left(x^2+x+1\right)^2}}$____$\frac{1\sqrt{3}}{9}t+\frac{1\sqrt{3}}{9}\sin t\cos t+C$$\frac{2\sqrt{3}}{9}t+\frac{2\sqrt{3}}{9}\sin t\cos t+C$$\frac{4\sqrt{3}}{9}t+\frac{4\sqrt{3}}{9}\sin t\cos t+C$$\frac{8\sqrt{3}}{9}t+\frac{8\sqrt{3}}{9}\sin t\cos t+C$
115113计算二次积分：$\int_0^{\pi}{\left(\int_x^{\pi}{\frac{\sin u}{u}\mathrm{d}u}\right)}\mathrm{d}x=$____0123
116114设$P_n\left(x\right)$是一个$n$次泰勒多项式，求$\int{\frac{P_n\left(x\right)}{\left(x-a\right)^{n+1}}}\mathrm{d}x$____$\sum_{k=0}^{n-1}{\frac{P_{n}^{\left( k \right)}\left( a \right)}{k!\left( k-n \right)}\cdot \left( x-a \right) ^{k-n}+\frac{P_{n}^{\left( n \right)}\left( a \right)}{n!}\cdot \ln \left| x-a \right|+C}$$\sum_{k=0}^{n-1}{\frac{P_{n}^{\left( k \right)}\left( a \right)}{(k-1)!\left( k-n \right)}\cdot \left( x-a \right) ^{k-n}+\frac{P_{n}^{\left( n \right)}\left( a \right)}{n!}\cdot \ln \left| x-a \right|+C}$$\sum_{k=0}^{n-1}{\frac{P_{n}^{\left( k \right)}\left( a \right)}{(k-1)!\left( k-n \right)}\cdot \left( x-a \right) ^{k-n}+\frac{P_{n}^{\left( n \right)}\left( a \right)}{(n+1)!}\cdot \ln \left| x-a \right|+C}$$\sum_{k=0}^{n-1}{\frac{P_{n}^{\left( k \right)}\left( a \right)}{k!\left( k-n \right)}\cdot \left( x-a \right) ^{k-n}+\frac{P_{n}^{\left( n \right)}\left( a \right)}{(n+1)!}\cdot \ln \left| x-a \right|+C}$
117115对于任意满足$f\left(1\right)=1$的$\left[0,1\right]$上的上凸函数$f\left(x\right)$，实数c满足：$\int_0^1{f^2\left(x\right)}\mathrm{d}x\geqslant c\int_0^1{f\left(x\right)}\mathrm{d}x$，求$c_{\max}$____$\frac{1}{3}$$\frac{2}{3}$1$\frac{4}{3}$
118116求级数$\sum_{n=1}^{\infty}{\frac{1}{n\left(n+1\right)\left(n+2\right)}}$的和____$\frac{1}{2}$$\frac{1}{4}$$\frac{1}{6}$$\frac{3}{8}$
119117计算定积分：$I=\int_{-2}^2{\frac{x^2+x}{\sqrt{4-x^2}}}\mathrm{d}x$____$\pi$$2\pi$$3\pi$$4\pi$
120118求极限：$\lim_{n\rightarrow\infty}\frac{5n+2\sqrt{n}+4}{\sqrt{n+1}}=$____-10${-\infty}$$+\infty$
121119函数项级数$\sum_{n=1}^{\infty}\left(\left({\frac{1}{5}}+{\frac{1}{n}}\right)^{n}+\left({\frac{1}{5}}\right)^{n}\right)x^{n}+\sum_{n=1}^{\infty}{\frac{\left({\frac{1}{5}}\right)^{n}}{n}}(x-2)^{n}$的收敛域是____$\left[-3,5\right]$$\left[-3,5\right)$$\left(-3,7\right]$$\left[-5,7\right]$
122120设$f\left(x\right)$在$\left[0,1\right]$上有二阶导数，且$f\prime\left(x\right)>0,f''\left(x\right)>0,f\left(0\right)=0$；取$x_1\in\left(0,1\right)$，数列$\left\{x_n\right\}$满足$\left(x_{n+1}-x_n\right)f\prime\left(x_n\right)+f\left(x_n\right)=0\left(n=1,2,\cdots\right)$；已知$\lim_{n\rightarrow\infty}x_n$存在，求其值____-101$\frac{1}{2}$
123121求不定积分：$\int{\left(1+x-\frac{1}{x}\right)\mathrm{e}^{x+\frac{1}{x}}}\mathrm{d}x$____$x\mathrm{e}^{x+\frac{1}{x}}+C$$x\mathrm{e}^{x+\frac{2}{x}}+C$$x\mathrm{e}^{x+\frac{1}{2x}}+C$$2x\mathrm{e}^{2x+\frac{1}{x}}+C$
124122计算广义积分：$I=\int_0^{+\infty}{\frac{\mathrm{cos}\ln x}{\left(1+x\right)^2}}\mathrm{d}x=$____$2\pi\text{csch}\pi$$\frac{\pi}{2}\text{csch}\pi$$\pi\text{csch}\pi$$2\pi\text{csch}\frac{\pi}{2}$
125123设$f\in C\left(\mathbb{R}^1\right)$，则$\int_0^{\frac{\pi}{2}}{\mathrm{d}\phi}\int_0^{\frac{\pi}{2}}{f\left(1-\sin\theta\cos\phi\right)}\sin\theta\mathrm{d}\theta=$____$\frac{\pi}{4}\int_0^1{f\left( x \right)}\mathrm{d}x$$\frac{\pi}{2}\int_0^1{2f\left( x \right)}\mathrm{d}x$$\frac{\pi}{2}\int_0^1{f\left( x \right)}\mathrm{d}x$$\pi\int_0^1{f\left( x \right)}\mathrm{d}x$
126124求极限：$\lim_{t\rightarrow0^+}\frac{\int_0^t{\mathrm{d}x}\int_x^t{\sin\left[\left(xy\right)^2\right]\mathrm{d}y}}{t^6}=$____$\frac{1}{6}$$\frac{1}{9}$$\frac{1}{12}$$\frac{1}{18}$
127125数列$\left\{x_n\right\}]$满足$x_1=1,x_2=\frac{1}{3}$，并且对于所有的$n\ge1$，满足$x_{n+2}=\frac{2x_nx_{n+1}}{x_n+x_{n+1}}$试求：$\lim_{n\rightarrow\infty}x_n=$____$\frac{3}{5}$$\frac{1}{2}$$\frac{3}{7}$$\frac{3}{8}$
128126设$D$为圆域$x^2+y^2\leq4x$，则$\iint_D{\mathrm{arc}\tan\left(\mathrm{e}^{xy}\right)}\mathrm{d}x\mathrm{d}y=$____$\frac{3\pi^{2}}{2}$$-\frac{\pi^{2}}{2}$$\frac{\pi^{2}}{2}$$\pi ^2$
129127求极限：$\lim_{t\rightarrow1^-}\sqrt{1-t}\left(1+t+t^4+t^9+\cdots+t^{n^2}+\cdots\right)=$____$\frac{\sqrt{\pi}}{2}$$\frac{\sqrt{\pi}}{3}$$\frac{\sqrt{\pi}}{4}$$\frac{\sqrt{\pi}}{5}$
130128求极限：$\lim_{n\rightarrow\infty}\left(\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+\cdots+\frac{1}{\sqrt{n^2+n}}\right)^n=$____$\text{e}^{-\frac{1}{5}}$$\text{e}^{-\frac{1}{6}}$$\text{e}^{-\frac{1}{3}}$$\text{e}^{-\frac{1}{4}}$
131129设二元函数$z=z\left(x,y\right)$是方程$z-{\frac{1}{9}}\sin z=4x+3y$所确定的隐函数，则$z_{x}=$____$\dfrac{-36}{9+\cos z}$$\dfrac{36}{9-\cos z}$$\frac{-36}{4+\cos z}$$\dfrac{36}{9+\cos z}$
132130计算定积分：$\int_{\frac{1}{\mathrm{e}}}^{\mathrm{e}}{\frac{\mathrm{arc}\tan\left(\ln x\right)}{x}}\mathrm{d}x$____01-1$\frac{2}{\pi}$
133131当$n\rightarrow\infty$时，$\frac{1}{3^n}\left(1+\frac{1}{n}\right)^{n^2}$的等价无穷小形式为${e}^{\alpha}\left(\beta\mathrm{e}\right)^n$，则$\alpha,\beta$为？____$\alpha=-\frac{1}{4},\beta=\frac{1}{3}$$\alpha=-\frac{1}{2},\beta=\frac{1}{3}$$\alpha=-\frac{1}{2},\beta=\frac{1}{4}$$\alpha=\frac{1}{4},\beta=\frac{1}{3}$
134132求不定积分：$I=\int{\frac{x\ln\left(x+\sqrt{1+x^2}\right)}{\left(1-x^2\right)^2}}\mathrm{d}x$____$\frac{\ln \left( x+\sqrt{1+x^2} \right)}{3\left( 1-x^2 \right)}-\frac{1}{4\sqrt{2}}\ln \left| \frac{\sqrt{x^2+1}+\sqrt{2}x}{\sqrt{x^2+1}-\sqrt{2}x} \right|+C$$\frac{\ln \left( x+\sqrt{1+x^2} \right)}{2\left( 1-x^2 \right)}-\frac{1}{4\sqrt{2}}\ln \left| \frac{\sqrt{x^2+1}+\sqrt{2}x}{\sqrt{x^2+1}-\sqrt{2}x} \right|+C$$\frac{\ln \left( x+\sqrt{1+x^2} \right)}{1-x^2}-\frac{1}{4\sqrt{2}}\ln \left| \frac{\sqrt{x^2+1}+\sqrt{2}x}{\sqrt{x^2+1}-\sqrt{2}x} \right|+C$$\frac{\ln \left( x+\sqrt{1+x^2} \right)}{2\left( 1-x^2 \right)}-\frac{1}{2\sqrt{2}}\ln \left| \frac{\sqrt{x^2+1}+\sqrt{2}x}{\sqrt{x^2+1}-\sqrt{2}x} \right|+C$
135133设$f\left(x\right)=\lim_{n\rightarrow\infty}\frac{x^{2n-1}+ax^2+bx}{x^{2n}+1}$为$\left(-\infty,+\infty\right)$上的连续函数，则a,b的值分别为？____a=0,b=1a=0,b=-1a=1,b=0a=-1,b=1
136134计算定积分：$I=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}{\sqrt{\sin^2x-\sin^4x}}\mathrm{d}x$____01-1\frac{1}{2}
137135求极限：$\lim_{x\rightarrow+\infty}\int_x^{2x}{\frac{\sqrt{t}}{1+\mathrm{e}^t}\mathrm{d}t}=$____-101$\frac{1}{2}$
138136试确定常数a和b使得极限$\lim_{x\rightarrow\infty}\left(\sqrt[3]{1-x^6}-ax^2-b\right)=0$成立____a=0,b=-1a=-1,b=0a=-1,b=1a=1,b=0
139137求极限：$\lim_{x\rightarrow2}\frac{x^2-4}{x^2-3x+2}$=____1234
140138求不定积分：$I=\int{\cos x\cdot\left(x^3+2x^2+3x+4\right)}\mathrm{d}x$____$\sin x\cdot \left( x^3+3x^2-3x \right) +\cos x\cdot \left( 3x^2+4x-3 \right) +C$$\sin x\cdot \left( x^3+3x^2-3x \right) +\cos x\cdot \left( 2x^2+4x-3 \right) +C$$\sin x\cdot \left( x^3+2x^2-3x \right) +\cos x\cdot \left( 3x^2+4x-3 \right) +C$$\sin x\cdot \left( x^3+2x^2-3x \right) +\cos x\cdot \left( 2x^2+4x-3 \right) +C$
141139求积分$I=\int_{0}^{\sqrt{5}\sin\frac{\pi}{3}}\mathrm{d}y\int_{\sqrt{5-y^{2}}}^{\sqrt{36-y^{2}}}\mathrm{d}x+\int_{\sqrt{5}\sin\frac{\pi}{3}}^{6\sin\frac{\pi}{3}}\mathrm{d}y\int_{y\cot\frac{\pi}{3}}^{\sqrt{36-y^{2}}}\mathrm{d}x=$____$\frac{31\pi}{18}$$\frac{31\pi}{6}$$\frac{31\pi}{15}$$\frac{31\pi}{9}$
142140设$f\left(x\right)$为连续函数，求极限：$\lim_{x\rightarrow0}\frac{\int_0^x{t\mathrm{e}^t}\left[\int_{t^2}^0{f\left(u\right)}\mathrm{d}u\right]\mathrm{d}t}{x^3\mathrm{e}^x}=$____-101$\frac{1}{2}$
143141求极限：$L=\lim_{n\rightarrow\infty}\int_0^1{\frac{\mathrm{d}x}{x^n+1}}$=____0123
144142给定以下数项级数(1)$\sum\limits_{n=1}^{\infty}\frac{(-1)^n}{n^{\frac{1}{4}}}\arctan\frac{1}{\sqrt{n}}$,(2)$\sum\limits_{n=1}^{\infty}(-1)^{\:n}\left(\dfrac{1}{\sqrt{n}}-\dfrac{1}{\sqrt{n+1}}\right)$,(3)$\sum\limits_{n=1}^{\infty}\sin\left(n\pi+\dfrac{1}{n}\pi\right)$,(4)$\sum\limits_{n=1}^\infty\left(\dfrac{\sin2n}{n^2}-\dfrac{1}{\sqrt{n}}\right)$,其中条件收敛的个数为____2341
145143求极限：$L=\lim_{x\rightarrow0}\frac{\ln^2\left(x+\sqrt{1+x^2}\right)+\mathrm{e}^{-x^2}-1}{x^4}=$____$\frac{1}{6}$$\frac{1}{4}$$\frac{1}{2}$$\frac{1}{3}$
146144求极限：$\lim_{n\rightarrow\infty}\left(\frac{2}{n^2+n+1}+\frac{4}{n^2+n+2}+\cdots+\frac{2n}{n^2+n+n}\right)^n$____$\mathrm{e}^{-\frac{3}{4}}$$\mathrm{e}^{-\frac{2}{3}}$$\mathrm{e}^{-\frac{3}{5}}$$\mathrm{e}^{-\frac{4}{5}}$
147145求极限：$L=\lim_{x\rightarrow0^+}\frac{\ln\left(\frac{\pi}{2}-\mathrm{arc}\tan\frac{1}{x\sqrt{1+x^2+x^4}}\right)}{\ln\left(\tan x\sqrt{2+x^2+x^4}\right)}=$____-101$\frac{1}{2}$
148146求极限：$\lim_{n\rightarrow\infty}\frac{5n+2\sqrt{n}+4}{\sqrt{n^3+1}}=$____0-112
149147设函数$f(x)\:=\:\sum_{n=0}^{\infty}{\frac{(-1)^{\:n}}{(2n+1)!}}\cdot{\frac{x^{2n+1}}{2^{2n}}}$，则$f^{(2022)}(1)\:=\:$____$\frac{1}{2^{2021}}\sin\left(\dfrac{1}{2}\right)$$-\frac{1}{2^{2022}}\sin\left(\frac{1}{2}\right)$$-\frac{1}{2^{2022}}\cos\left(\frac{1}{2}\right)$$-\frac{1}{2^{2021}}\sin\left(\frac{1}{2}\right)$
150148设函数$z=z\left(x,y\right)$由方程$F\left(\frac{y}{x},\frac{z}{x}\right)=0$确定，其中$F$为可微函数，且$F_{2}^{\prime}\ne0$，则$x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}=$____z1-2z0
151149求不定积分：$\int{\frac{x\sqrt[3]{\mathrm{arc}\tan\left(x^2\right)}}{1+x^4}}\mathrm{d}x=$____$\frac{3}{8}\left[ \mathrm{arc}\tan \left( x^2 \right) \right] ^{\frac{4}{3}}+C$$\frac{1}{2}\left[ \mathrm{arc}\tan \left( x^2 \right) \right] ^{\frac{4}{3}}+C$$\frac{5}{8}\left[ \mathrm{arc}\tan \left( x^2 \right) \right] ^{\frac{4}{3}}+C$$\frac{3}{4}\left[ \mathrm{arc}\tan \left( x^2 \right) \right] ^{\frac{4}{3}}+C$
152150设$\Omega$是由曲线$\left\{\begin{array}{l} y^2=2z\\ x=0\\ \end{array}\right.$绕$z$轴旋转一周而成的曲面与$z=2,z=8$所围立体；计算$\iiint_{\Omega}{\left(x^2+y^2\right)\mathrm{d}V}=$____$330\pi$$336\pi$$342\pi$$348\pi$
153151求定积分：$\int_0^1{\sqrt{2x-x^2}}\mathrm{d}x=$____$\frac{\pi}{4}$$\frac{\pi}{2}$$\frac{3\pi}{4}$$\pi$
154152求极限：$\lim_{x\rightarrow0}\left(\frac{\sin\sin x}{\sin\mathrm{arc}\tan x}\right)^{\frac{1}{1-\cos x}}=$____$\sqrt[3]{e}$$\sqrt[3]{2e}$$\sqrt[3]{3e}$$\sqrt[3]{4e}$
155153设$f\left(x\right)$在$\left[-1,1\right]$上有定义，$f\prime\left(0\right)$存在，求极限：$\lim_{n\rightarrow\infty}\left[f\left(\frac{1}{n^2}\right)+f\left(\frac{2}{n^2}\right)+\cdots+f\left(\frac{n}{n^2}\right)-nf\left(0\right)\right]=$____$\dfrac{1}{3}f'(0)$$\dfrac{1}{2}f'(0)$$\dfrac{1}{4}f'(0)$$\dfrac{2}{3}f'(0)$
156154判断广义积分$\int_1^{+\infty}{\frac{\sqrt{1+x^{-1}}-1}{x^p\ln\left(1+x^{-2}\right)}}\mathrm{d}x$的敛散性____$\left\{\begin{array}{c}p\leq1\text{时发散}\\ p\gt1 \text{时收敛}\end{array}\right.$$\left\{\begin{array}{c}p\leq2\text{时发散}\\ p\gt2 \text{时收敛}\end{array}\right.$$\left\{\begin{array}{c}p\leq3\text{时发散}\\ p\gt3 \text{时收敛}\end{array}\right.$$\left\{\begin{array}{c}p\leq3\text{时发散}\\ p\gt3 \text{时收敛}\end{array}\right.$
157155求极限：$\lim_{x\rightarrow0}\frac{\int_0^x{\left|\sin t\right|}\mathrm{d}t}{\int_0^x{\left(t-\left[t\right]\right)}\mathrm{d}t}$____$\frac{2}{\pi}$$\frac{4}{\pi}$$\frac{6}{\pi}$$\frac{8}{\pi}$
158156求不定积分：$I=\int{\frac{1+\sin^2x}{1-\cos2x}}\mathrm{d}x$____$-\frac{1}{3}\cot x+\frac{1}{2}x+C$$-\frac{1}{2}\cot x+\frac{1}{3}x+C$$-\frac{1}{2}\cot x+\frac{1}{2}x+C$$-\frac{1}{2}\tan x+\frac{1}{2}x+C$
159157求不定积分：$\int{\frac{x^2}{\sqrt{3+2x-x^2}}}\mathrm{d}x$____$6\mathrm{arc}\sin \left( \frac{x-1}{2} \right) -\frac{\sqrt{3+2x-x^2}}{2}\cdot \left( x+3 \right) +C$$3\mathrm{arc}\sin \left( \frac{x-1}{3} \right) -\frac{\sqrt{3+2x-x^2}}{2}\cdot \left( x+3 \right) +C$$3\mathrm{arc}\sin \left( \frac{x-1}{2} \right) -\frac{\sqrt{3+2x-x^2}}{2}\cdot \left( x+3 \right) +C$$6\mathrm{arc}\sin \left( \frac{x-1}{2} \right) -\frac{\sqrt{3+2x-x^2}}{2}\cdot \left( x+6 \right) +C$
160158求不定积分：$\int{\frac{x}{\left(x-1\right)\left(x^2+1\right)}}\mathrm{d}x$____$\frac{1}{4}\mathrm{arc}\tan x+\frac{1}{4}\ln \left| \frac{x-1}{x+1} \right|+\frac{1}{4}\ln \left| \frac{x^2-1}{x^2+1} \right|+C$$\frac{1}{4}\mathrm{arc}\tan x+\frac{1}{2}\ln \left| \frac{x-1}{x+1} \right|+\frac{1}{4}\ln \left| \frac{x^2-1}{x^2+1} \right|+C$$\frac{1}{2}\mathrm{arc}\tan x+\frac{1}{4}\ln \left| \frac{x-1}{x+1} \right|+\frac{1}{4}\ln \left| \frac{x^2-1}{x^2+1} \right|+C$$\frac{1}{4}\mathrm{arc}\tan x+\frac{1}{2}\ln \left| \frac{x-1}{x+1} \right|+\frac{1}{2}\ln \left| \frac{x^2-1}{x^2+1} \right|+C$
161159设$L$为圆周$x^2+y^2=2x$，计算曲线积分：$I=\int_L{x\mathrm{d}s}=$____$\pi$$2\pi$$3\pi$$4\pi$
162160已知曲线$\Gamma:4x^{2}+2y^{2}=1$，$\boldsymbol{n}$是$\Gamma$在点$\mathrm{M}\left({\sqrt{\frac{1}{8}}},{\sqrt{\frac{1}{4}}}\right)$处的一个外法向量（指向曲线$\Gamma$所围区域的外部的方向），函数$z=3x-(4x^{2}+2y^{2})$，则方向导数$\left.\frac{\partial z}{\partial\boldsymbol{n}}\right|_{_{M}}$=____$\sqrt{6}-\sqrt{3}$$\sqrt{6}$$\sqrt{6}-2\sqrt{3}$$-\sqrt{3}$
163161求不定积分：$I=\int{\mathrm{arc}\tan\sqrt{\frac{1-x}{1+x}}\mathrm{d}x}$____$x\mathrm{arc}\tan \sqrt{\frac{2-x}{2+x}}-\frac{1}{2}\sqrt{1-x^2}+C$$x\mathrm{arc}\tan \sqrt{\frac{2-x}{2+x}}-\frac{1}{3}\sqrt{1-x^2}+C$$x\mathrm{arc}\tan \sqrt{\frac{2-x}{1+x}}-\frac{2}{3}\sqrt{1-x^2}+C$$x\mathrm{arc}\tan \sqrt{\frac{1-x}{1+x}}-\frac{1}{2}\sqrt{1-x^2}+C$
164162求极限：$\lim_{n\rightarrow\infty}\sum_{k=1}^n{\frac{n+1-k}{nC_{n}^{k}}}$____0-11$\dfrac{1}{2}$
165163求不定积分：$I=\int{\sqrt{x^2-1}}\mathrm{d}x$=____$\frac{x}{4}\sqrt{x^{2}-1}-\frac{1}{4}\operatorname{ln}\big|x+\sqrt{x^{2}-1}\big|+C$$\frac{x}{4}\sqrt{x^{2}-1}-\frac{1}{2}\operatorname{ln}\big|x+\sqrt{x^{2}-1}\big|+C$$\frac{x}{2}\sqrt{x^{2}-1}-\frac{1}{4}\operatorname{ln}\big|x+\sqrt{x^{2}-1}\big|+C$$\frac{x}{2}\sqrt{x^{2}-1}-\frac{1}{2}\operatorname{ln}\big|x+\sqrt{x^{2}-1}\big|+C$
166164已知$a_n$收敛，且设$\operatorname*{lim}_{n\to\infty}a_{n}=A$求极限：$\lim_{n\rightarrow\infty}\frac{\sum_{k=1}^n{k^ma_k}}{n^{m+1}}=$____$\frac{A}{m-1}$$\frac{A}{m}$$\frac{A}{m+1}$$\frac{2A}{m}$
167165求极限：$\lim_{n\rightarrow\infty}\sum_{k=1}^n{\frac{k}{n^2+k}}\sin^2\frac{\pi\left(k-1\right)}{n}$=____$\begin{aligned}\frac{1}{2}\left(1+\sin^21-\sin2\right)\end{aligned}$$\begin{aligned}\frac{1}{3}\left(1+\sin^21-\sin2\right)\end{aligned}$$\begin{aligned}\frac{1}{2}\left(1+\sin^22-\sin2\right)\end{aligned}$$\begin{aligned}\frac{1}{4}\left(1+\sin^21-\sin2\right)\end{aligned}$
168166求不定积分：$\int{\frac{x-\sin x\cos x}{x^2\cos^2x+\sin^2x}}$____$-\mathrm{arc}\tan \left( x\cos x \right) +C$$-\mathrm{arc}\tan \left( x\sin x \right) +C$$-\mathrm{arc}\tan \left( x\tan x \right) +C$$-\mathrm{arc}\tan \left( x\cot x \right) +C$
169167设$D=\left\{\left(x,y\right)|x^2+y^2\leq4\right\}$，则$\iint_D{\left(x+2y\right)^2\mathrm{d}x\mathrm{d}y}=$____$10\pi$$20\pi$$40\pi$$50\pi$
170168求不定积分：$\int{\frac{x^2}{\sqrt{a^2+x^2}}}\mathrm{d}x$=____$\dfrac{x}{2}\sqrt{x^2+a^2}-\dfrac{a^2}{2}\ln\left|x+\sqrt{x^2+a^2}\right|+C$$\dfrac{x}{3}\sqrt{x^2+a^2}-\dfrac{a^2}{2}\ln\left|x+\sqrt{x^2+a^2}\right|+C$$\dfrac{x}{2}\sqrt{x^2+a^2}-\dfrac{a^2}{3}\ln\left|x+\sqrt{x^2+a^2}\right|+C$$\dfrac{2x}{3}\sqrt{x^2+a^2}-\dfrac{2a^2}{3}\ln\left|x+\sqrt{x^2+a^2}\right|+C$
171169求极限：$\lim_{n\rightarrow\infty}\frac{1}{n^2}\sum_{j=1}^{\left[\mathrm{e}^n\right]}{\sum_{i=3n}^{7n}{\sin\left(\frac{i}{n}\cdot\frac{j}{n}\right)}}$____$\ln 7-\ln 3$$2\ln 2-\ln 3$$\ln 3$$\ln 10-\ln 3$
172170利用泰勒展开求解极限：$\lim_{x\rightarrow0}\frac{\tan x\cdot\mathrm{arc}\tan x-x^2}{x^6}=$____$\frac{4}{7}$$\frac{4}{9}$$\frac{1}{3}$$\frac{2}{9}$
173171定义：$f\left(\alpha\right)=\int_0^{\alpha}{\left[\ln x\ln\left(\alpha-x\right)\right]}\mathrm{d}x$，求出$\alpha$的值，使得$f\left(\alpha\right)$最小____$\mathrm{e}^{\frac{\pi}{\sqrt{12}}}$$\mathrm{e}^{\frac{\pi}{\sqrt{6}}}$$\mathrm{e}^{\frac{\pi}{\sqrt{3}}}$$\mathrm{e}^{2\frac{\pi}{\sqrt{3}}}$
174172求极限：$\lim_{x\rightarrow0}\frac{\overset{p\text{次}}{\overbrace{\tan\tan\cdots\tan}}x-\overset{p\text{次}}{\overbrace{\sin\sin\cdots\sin}x}}{\tan x-\sin x}=$，其中$p\in\mathbb{N}^+$____p-1pp+12p