2.应用格林公式计算下列曲线所围的平面面积:-|||-(1)星形线: =a(cos )^3t, =a(sin )^3t;-|||-(2)双纽线: (({x)^2+(y)^2)}^2=(a)^2((x)^2-(y)^2).

设在某一规定的时间间隔里,某电气设备用于最大负荷的-|||-时间X(以min计)是一个随机变量,其概率密度为-|||-leqslant xleqslant 1500,-|||-f(x)= ^2)x, dfrac (1)({(1500))^2}(x-3000) . , lt xleqslant 3000,-|||-其他.-|||-求E(X).

求下列向量组的秩及一个极大无关组,并将不属于极大无关组的向量由极大无关-|||-组线性表示:-|||-(1) 1 -3 -1 1)-|||-α1= 3 α2= 2 α3= -3 α4= 4 α5= 5-|||-1 2 -9 -8 -1

.设函数-|||-f(x)= { ,xneq 0, 0,x=0. . ,(m为正整数),-|||-试问:(1)m等于何值时,f在 x=0 连续;-|||-(2)m等于何值时,f在 x=0 可导。

[例5.14]求微分方程 y''-5y'+6y=0 的通解.

已知非齐次线性方程组 ) (x)_(1)+(x)_(2)+(x)_(3)+(x)_(4)=-1 4(x)_(1)+3(x)_(2)+5(x)_(3)-(x)_(4)=-1 a(x)_(1)+(x)_(2)+3(x)_(3) . 有三个线性无关的解。 (Ⅰ)证明方程组系数矩阵A的秩r(A) =2； (Ⅱ)求a，b的值及方程组的通解．

一、判断题(对的在括号中打"√",错的在括号中打"×")-|||-1.若f(x)为(a,b)内的单调函数,f(x)在(a,b)内可导,则 '(x)gt 0. ()-|||-2.单调可导函数的导函数必定单调. ()-|||-3.若 ^m((x)_(0))=0, 则(x0,f(x0))必为曲线 y=f(x) 的拐点, ()-|||-4.若(x0,f(x0))为曲线 y=f(x) 的拐点,则有 ^m((x)_(0))=0. ()-|||-5.若(x0,f(x0))为连续曲线 y=f(x) 上的凹弧与凸弧的分界点,则(x0,f(x0 ))为该曲线-|||-()-|||-的拐点.

求下列不定积分:-|||-(22) int dfrac (cos 2x)({cos )^2x(sin )^2x}dx ;

2.求下列不定积分(其中a,b,w,φ均为常数):-|||-(1)int (e)^5tdt;-|||-(2) int ((3-2x))^3dx ;-|||-(3) int dfrac (dx)(1-2x) =-|||-(4) int dfrac (dx)(sqrt [3]{2-3x)} :-|||-(5) int (sin ax-(e)^dfrac (x{b)})dx ;-|||-(6) int dfrac (sin sqrt {t)}(sqrt {t)}dt =-|||-(7) int x(e)^-(x^2)dx ;-|||-(8)int xcos ((x)^2)dx;-|||-(9) int dfrac (x)(sqrt {2-3{x)^2}}dx :-|||-(10) int dfrac (3{x)^3}(1-{x)^4}dx :-|||-(11) int dfrac (x+1)({x)^2+2x+5}dx ;-|||-(12) int (cos )^2(omega t+varphi )sin (omega t+varphi )dt ;-|||-(13) int dfrac (sin x)({cos )^3x}dx =-|||-(14) int dfrac (sin x+cos x)(sqrt [3]{sin x-cos x)}dx :-|||-(15)int (tan )^10xcdot (sec )^2xdx;-|||-(16) int dfrac (dx)(xln xln ln x)-|||-(17) int dfrac (dx)({(arcsin x))^2sqrt (1-{x)^2}} :-|||-__-|||-(18) int dfrac ({10)^2arcsin x}(sqrt {1-{x)^2}}dx :

计算定积分 (int )_(0)^1(2x+1)dx.

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