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9.7.1 无穷积分的符号计算及其MATLAB程序

9.7.1  无穷积分的符号计算及其MATLAB程序

例9.7.1  讨论反常积分d的敛散性.

解 输入程序

>> syms x

F1=int((5*x)/(x^4+2),x,1,+inf) LimF1= double(F1)

F2=int((5*x.^2)/(x^4+2),x,1,+inf), LimF2= double(F2)

F3=int((5*x.^3)/(x^4+2),x,1,+inf), LimF3= double(F3)

F8=int((5*x.^8)/(x^4+2),x,1,+inf), LimF8= double(F8)

运行后屏幕显示如下

F1 =

5/8*pi*2^(1/2)-5/4*atan(1/2*2^(1/2))*2^(1/2)

LimF1 =

1.6888

F2=

5/4*2^(1/4)*pi-5/8*2^(1/4)*log(1-2^(3/4)+2^(1/2))+5/8*2^(1/4)*log(1+2^(3/4)+2^(1/2))-5/4*2^(1/4)*atan(2^(1/4)+1)-5/4*2^(1/4)*atan(2^(1/4)-1)

LimF2 =       F3 =     LimF3 =       F8 =       LimF8 =

3.9734       inf          Inf        inf           Inf

 

例9.7.2  讨论反常积分d的敛散性.

解 当时,输入程序

>> syms x

Ff2=int(1/x^(-2*2),x, -inf,-1),

F02=int(1/x^(2*0.2),x, -inf,-1)

LimF02= double(F02), F05=int(1/x^(0.5*2),x, -inf,-1)

F1=int(1/x^(1*2),x, -inf,-1), F4=int(1/x^(4*2),x, -inf,-1)

运行后屏幕显示

Ff2 =       F02 =                    LimF02 =           F05 =

Inf         -(-1)^(3/5)*inf            Inf – Infi       -inf

F1 =      F4 =

1         1/7

 

例9.7.3  讨论反常积分d)的敛散性.

解 输入程序如下

>>  syms x

F1=int(1/(x^2+2*x+3),x,-inf, +inf), LimF1= double(F1)

F2=int(1/(x^2+2*x-3),x, -inf,0), LimF2= double(F2)

F3=int(1/(x^2+2*x-3),x, 0,+inf), LimF3= double(F3)

F=F2+F3,LimF=LimF2+ LimF3

运行后屏幕显示如下

F1 =                   LimF1 =

1/2*pi*2^(1/2)             2.221441469079183e+000

F2 =         LimF2 =            F3 =           LimF3 =

NaN              NaN              NaN               NaN

F =          LimF =

NaN              NaN

即 d收敛,而d不存在,发散.为什么呢?看了被积函数的图形就明白了.

输入程序

>> subplot(1,2,1),

x =-10: 0.01:10; y=1./(x.^2+2*x+3);

plot(x,y), grid ,title('y=1/(x^2+2x+3)')

subplot(1,2,2),

x =-10: 0.01:10; %z=1./(x.^2+2*x-3);

fplot('1./(x.^2+2*x-3)',[-10 10 -2 2]),

grid,title(' y=1/(x^2+2x-3)')

运行后屏幕显示图形(略).

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