查看“︁Maple/积分”︁的源代码


Maple 积分符号

Int(x^2,x);

显示  求<math>\int x^2dx</math>；

int(x^2,x);

<math>\frac{1}{3} x^3</math>；

==简单积分==
f:=8x^3+25x^2-3x-1;

int(f,x)

<math>2*x^4+(25/3)*x^3-(3/2)*x^2-x</math>

int(1/x,x);
<math>ln(x)</math>

int(a^x,x);

<math>\frac{a^x}{ln(a)}</math>

int(sin(x),x)

<math>-\cos(x)</math>

int(sqrt{x^2+a^2},x);


<math>(1/2)*x*\sqrt(x^2+a^2)+(1/2)*a^2*ln(x+\sqrt(x^2+a^2))</math>

int(sqrt(x^2+a^2)/x,x);

<math>\sqrt(x^2+a^2)-a^2*ln((2*a^2+2*\sqrt(a^2)*\sqrt(x^2+a^2))/x)/\sqrt(a^2)</math>

int(sqrt(x^2+a^2)/x^5,x);

 

<math>-(1/4)*(x^2+a^2)^(3/2)/(a^2*x^4)+</math>
<math>  (1/8)*(x^2+a^2)^(3/2)/(a^4*x^2)-</math>
<math>  (1/8)*\sqrt(x^2+a^2)/a^4+</math>
<math>  (1/8)*ln((2*a^2+2*\sqrt(a^2)*sqrt(x^2+a^2))/x)/(a^2*\sqrt(a^2))</math>

int((x^2+a^2)^3/2,x)

<math>(1/14)*x^7+(3/10)*a^2*x^5+(1/2)*a^4*x^3+(1/2)*a^6*x</math>

==复杂积分==
int(ln(a^(exp(x^2))), x)；

<math>x*\ln(a^(exp(x^2)))-x*\exp(x^2)*ln(a)-(1/2*I)*\ln(a)*\sqrt(Pi)*erf(I*x)></math>

int(cos(tan(x)), x)

<math>-(1/2*I)*(-I*Si(tan(x)-I)*sinh(1)</math>
<math>+Ci(tan(x)-I)*cosh(1))</math>
<math>+(1/2*I)*(I*Si(tan(x)+I)*sinh(1)</math>
<math>+Ci(tan(x)+I)*cosh(1))</math>

int(cos(x^3), x);

(1/6)*sqrt(Pi)*2^(1/3)*((9/2)*2^(2/3)*((2/7)*x^6+2/3)*sin(x^3)/(sqrt(Pi)*x^2)+3*2^(2/3)*(cos(x^3)*x^3-sin(x^3))/(sqrt(Pi)*x^2)-(9/7)*x^7*2^(2/3)*sin(x^3)*LommelS1(11/6, 3/2, x^3)/(sqrt(Pi)*(x^3)^(11/6))-3*x^7*2^(2/3)*(cos(x^3)*x^3-sin(x^3))*LommelS1(5/6, 1/2, x^3)/(sqrt(Pi)*(x^3)^(17/6)))

int(exp(tan(x)), x)

(1/2*I)*exp(I)*Ei(1, -tan(x)+I)-(1/2*I)*exp(-I)*Ei(1, -tan(x)-I)

int(ln(sin(x)*cos(x)), x)

(2*I)*ln(2)*ln(exp(I*x))-I*ln(exp(I*x))*ln(I*(-(exp(I*x))^2-1)*((exp(I*x))^2-1)/(exp(I*x))^2)-I*dilog(exp(I*x))+I*dilog(exp(I*x)+1)+I*ln(exp(I*x))*ln(exp(I*x)+1)-I*ln(exp(I*x))^2+I*ln(exp(I*x))*ln(1+I*exp(I*x))+I*ln(exp(I*x))*ln(1-I*exp(I*x))+I*dilog(1+I*exp(I*x))+I*dilog(1-I*exp(I*x))

int(sin(x^(1/3)), x)

-3*x^(2/3)*cos(x^(1/3))+6*cos(x^(1/3))+6*x^(1/3)*sin(x^(1/3))

int(x^3*ln(sin(x)), x)

-(1/4)*x^4*ln(exp(I*x))+(1/4)*x^4*ln(exp((2*I)*x)-1)-(1/4)*x^4*ln(1-exp(I*x))+I*x^3*polylog(2, exp(I*x))-3*x^2*polylog(3, exp(I*x))+(1/8*I)*x^4*Pi*csgn(sin(x))^3+6*polylog(5, exp(I*x))-(1/4)*x^4*ln(exp(I*x)+1)+(1/8*I)*x^4*Pi*csgn(I*sin(x))*csgn(sin(x))-3*x^2*polylog(3, -exp(I*x))+(1/8*I)*x^4*Pi*csgn(sin(x))*csgn(I*exp(-I*x))*csgn(I*(exp((2*I)*x)-1))+6*polylog(5, -exp(I*x))+(1/4)*x^4*ln(-1/2*I)+I*x^3*polylog(2, -exp(I*x))+(1/8*I)*x^4*Pi*csgn(sin(x))^2*csgn(I*exp(-I*x))-(6*I)*x*polylog(4, -exp(I*x))+(1/8*I)*x^4*Pi*csgn(I*sin(x))^2+(1/20*I)*x^5-(1/8*I)*x^4*Pi*csgn(I*sin(x))^3-(6*I)*x*polylog(4, exp(I*x))+(1/8*I)*x^4*Pi*csgn(sin(x))^2*csgn(I*(exp((2*I)*x)-1))-(1/8*I)*x^4*Pi*csgn(I*sin(x))^2*csgn(sin(x))
==重积分==
int(x^2*cos(x^2), x, x)

<math>\int\int(x^2*cos(x^2)dxdx</math>

-(1/4)*cos(x^2)-(1/4)*Pi*(FresnelS(sqrt(2)*x/sqrt(Pi))*sqrt(2)*x/sqrt(Pi)+cos(x^2)/Pi)


Int(ln(tan(x)*cos(y)), x, y)

<math>\int\int(ln(tan(x)*cos(y))dxdy</math>


int(ln(tan(x)*cos(y)), x, y)

 1   /cos(x) + I sin(x)\                      1   /
 - ln|-----------------| ln(2) ln(exp(I y)) - - ln|
 2   \     cos(x)      /                      2   \

                                     /       /          2    \\
   cos(x) + I sin(x)\                |sin(x) \(exp(I y))  + 1/|
   -----------------| ln(exp(I y)) ln|------------------------|
        cos(x)      /                \    cos(x) exp(I y)     /

      1   /cos(x) + I sin(x)\             2
    - - ln|-----------------| ln(exp(I y)) 
      4   \     cos(x)      /              

      1   /cos(x) + I sin(x)\                                
    + - ln|-----------------| ln(exp(I y)) ln(1 + I exp(I y))
      2   \     cos(x)      /                                

      1   /cos(x) + I sin(x)\                                
    + - ln|-----------------| ln(exp(I y)) ln(1 - I exp(I y))
      2   \     cos(x)      /                                

      1   /cos(x) + I sin(x)\                      
    + - ln|-----------------| dilog(1 + I exp(I y))
      2   \     cos(x)      /                      

      1   /cos(x) + I sin(x)\                      
    + - ln|-----------------| dilog(1 - I exp(I y))
      2   \     cos(x)      /                      

      1   /cos(x) - I sin(x)\                      1   /
    - - ln|-----------------| ln(2) ln(exp(I y)) + - ln|
      2   \     cos(x)      /                      2   \

                                     /       /          2    \\
   cos(x) - I sin(x)\                |sin(x) \(exp(I y))  + 1/|
   -----------------| ln(exp(I y)) ln|------------------------|
        cos(x)      /                \    cos(x) exp(I y)     /

      1   /cos(x) - I sin(x)\             2
    + - ln|-----------------| ln(exp(I y)) 
      4   \     cos(x)      /              

      1   /cos(x) - I sin(x)\                                
    - - ln|-----------------| ln(exp(I y)) ln(1 + I exp(I y))
      2   \     cos(x)      /                                

      1   /cos(x) - I sin(x)\                                
    - - ln|-----------------| ln(exp(I y)) ln(1 - I exp(I y))
      2   \     cos(x)      /                                

      1   /cos(x) - I sin(x)\                      
    - - ln|-----------------| dilog(1 + I exp(I y))
      2   \     cos(x)      /                      

      1   /cos(x) - I sin(x)\                      
    - - ln|-----------------| dilog(1 - I exp(I y))
      2   \     cos(x)      /                      

      1                           1                        
    - - I dilog(1 + I tan(x)) y + - I dilog(1 - I tan(x)) y
      2                           2                        



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