用拉格朗日中值定理证明不等式：(x)/(1+x)＜ln（1+x）＜x（x＞0）．

24. int (e)^sqrt (3x+9)dx.

(2) (x)^n-((x'))^2+((x'))^3=0-|||-(3) ^n+dfrac (2)(1-x)((x'))^2=0 ;-|||-(4) ''+sqrt (1-{(x'))^2}=0 ;-|||-(5) (x)^11+([ 1+{(x'))^2] }^3/2=0 (常数 neq 0) ;-|||-(6) ''-dfrac (1)(t)x'+((x'))^2=0 (提示:方程两端除以x').

[题目]设 '((x)_(0))=f'((x)_(0))=0 ,f``(x0)>0, 则下列选项-|||-正确的是 ()-|||-A.f(x0)是f`(x)的极大值-|||-B.f(x0)是f(x)的极大值-|||-C.f(x0)是f(x)极小值-|||-D.(x0,f(x0))是曲线 y=f(x) 的拐点

设A为n阶对称矩阵,B为n阶反对称矩阵,证明;(1)AB-BA为对称矩阵;(2)AB+BA为反对称矩阵。

(19) int tan sqrt (1+{x)^2}cdot dfrac (xdx)(sqrt {1+{x)^2}}-|||-(20) int dfrac (arctan sqrt {x)}(sqrt {x)(1+x)}dx-|||-21 f(1thx)/(t+sx)^2dx;-|||-(22) int dfrac (dx)(sin xcos x)-|||-(23) int dfrac (ln tan x)(cos xsin x)dx-|||-(24)|cosx|xdx-|||-(25) int (cos )^2(omega t+varphi )dt;-|||-circled (19)int sin 2xcos 3xdx;-|||-(27) int cos xcos dfrac (x)(2)dx;-|||-(28)∫sin 5xsin77xdx;-|||-(29)∫tan^2 xsee xdx;-|||-(30) int dfrac (dx)({e)^x+(e)^-x}-|||-(31) int dfrac (1-x)(sqrt {9-4{x)^2}}dx;-|||-(32) int dfrac ({x)^3}(9+{x)^2}dx;-|||-(33) int dfrac (dx)(2{x)^2-1}-|||-34 f(4h)/((x+1(x-2))^2-|||-(35) int dfrac (x)({x)^2-x-2}dx;-|||-36 int dfrac ({x)^2dx}(sqrt {{a)^2-(x)^2}}(agt 0)-|||-17 int dfrac (dx)(xsqrt {{x)^2-1}}-|||-(38) int dfrac (dx)(sqrt {{({x)^2+1)}^x}}-|||-(39) int dfrac (sqrt {{x)^2-9}}(x)dx-|||-40] dfrac (dx)(1+sqrt {2x)}-|||-(41) int dfrac (dx)(1+sqrt {1-{x)^2}}-|||-(42) int dfrac (dx)(x+sqrt {1-{x)^2}}-|||-(43) int dfrac (x-1)({x)^2+2x+3}dx;-|||-(44) int dfrac ({x)^3+1}({({x)^2+1)}^2}dx-|||-前面我们在复合函数求导法则的基础上,得到了换元积分法.现在我-|||-两个函数乘积的求导法则,来推得另一个求积分的基本方法一 爸想-|||-设函数 u=u(x) 及 v=v(x) 具有连续导数,则两个函数乘积的导数化-|||-(uv)'=u'w+uv'-|||-(uv)'=u'w+uv'-|||-移项,得-|||-'=((uv))^t-(u)^vv.-|||-对这个等式两边求不定积分,得-|||-f uv`dx=uv- ∫u`vdx

2.4设(X,Ⅱ·∥)是赋范线性空间,对于 ,yin X 令-|||-d(x,y)= ) 0, x=y, ||x-y||+1,xneq y, .-|||-证明:d是距离,但不能由某个范数导出,即不存在X上的范数∥·|,使得-|||-d(x,y)=||x-y||,, x,y∈X.

6.设mE<∞，f_{n)(x)}为a.e.有限可测函数列，证明：lim_(ntoinfty)int_(E)(|f_(n)(x)|)/(1+|f_(n)(x)|)dx=0的充要条件是f_(n)Rightarrow0.

1.(1)在下列句子中随机地取一个单词,以X表示取到的单词所包含的字母个数,写出-|||-X的分布律并求E (X).-|||-THE GIRL PUT ON HE R BEAUTIFUL R ED HAT".

3.求下列齐次线性方程组的基础解系，并给出其结构式通解.(1) 2x_(1)+3x_(2)-x_(3)-5x_(4)=0,(2)}2x_(1)+x_(2)-3x_(3)+2x_(4)=0,3x_(1)+2x_(2)+x_(3)-2x_(4)=0,x_(1)+x_(2)+4x_(3)-4x_(4)=0;(3)}x_(1)+2x_(2)+7x_(4)-4x_(5)=0,x_(1)-x_(2)+3x_(3)-2x_(4)-x_(5)=0,2x_(1)+4x_(3)+2x_(4)-4x_(5)=0,x_(1)+x_(2)+x_(3)+4x_(4)-3x_(5)=0;(4)}x_(1)+2x_(2)-3x_(3)=0,2x_(1)+x_(2)+2x_(3)=0,x_(1)-x_(2)+3x_(3)=0.