Maple/積分

Maple 積分符號

Int(x^2,x);

顯示 求$\int x^{2}dx$；

int(x^2,x);

$\frac{1}{3}{x}^3$；

簡單積分

f:=8x^3+25x^2-3x-1;

int(f,x)

$2\ast x^4+(25/3)\ast x^3-(3/2)\ast x^2-x$

int(1/x,x); $ln(x)$

int(a^x,x);

$\frac{a^x}{ln(a)}$

int(sin(x),x)

$-\cos (x)$

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

$(1/2)\ast x\ast \sqrt{(}{x}^2+a^2)+(1/2)\ast a^2\ast ln(x+\sqrt{(}{x}^2+a^2))$

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

$\sqrt{(}{x}^2+a^2)-a^2\ast ln((2\ast a^2+2\ast \sqrt{(}{a}^2)\ast \sqrt{(}{x}^2+a^2))/x)/\sqrt{(}{a}^2)$

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

$-(1/4)\ast (x^2+a^2{)}^{(}3/2)/(a^2\ast x^4)+$ $(1/8)\ast (x^2+a^2{)}^{(}3/2)/(a^4\ast x^2)-$ $(1/8)\ast \sqrt{(}{x}^2+a^2)/a^4+$ $(1/8)\ast ln((2\ast a^2+2\ast \sqrt{(}{a}^2)\ast sqrt(x^2+a^2))/x)/(a^2\ast \sqrt{(}{a}^2))$

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

$(1/14)\ast x^7+(3/10)\ast a^2\ast x^5+(1/2)\ast a^4\ast x^3+(1/2)\ast a^6\ast x$

複雜積分

int(ln(a^(exp(x^2))), x)；

$x\ast \ln (a^{(}exp(x^2)))-x\ast \exp (x^2)\ast ln(a)-(1/2\ast I)\ast \ln (a)\ast \sqrt{(}Pi)\ast erf(I\ast x)>$

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

$-(1/2\ast I)\ast (-I\ast Si(tan(x)-I)\ast sinh(1)$ $+Ci(tan(x)-I)\ast cosh(1))$ $+(1/2\ast I)\ast (I\ast Si(tan(x)+I)\ast sinh(1)$ $+Ci(tan(x)+I)\ast cosh(1))$

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))

重積分

int(x^2*cos(x^2), x, x)

$\int \int (x^2\ast cos(x^2)dxdx$

-(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)

$\int \int (ln(tan(x)\ast cos(y))dxdy$

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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