# Maple/积分

Maple 积分符号

Int(x^2,x);

int(x^2,x);

${\displaystyle {\frac {1}{3}}x^{3}}$

## 简单积分

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

int(f,x)

${\displaystyle 2*x^{4}+(25/3)*x^{3}-(3/2)*x^{2}-x}$

int(1/x,x); ${\displaystyle ln(x)}$

int(a^x,x);

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

int(sin(x),x)

${\displaystyle -\cos(x)}$

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

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

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

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

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

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

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

${\displaystyle (1/14)*x^{7}+(3/10)*a^{2}*x^{5}+(1/2)*a^{4}*x^{3}+(1/2)*a^{6}*x}$

## 复杂积分

int(ln(a^(exp(x^2))), x)；

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

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

${\displaystyle -(1/2*I)*(-I*Si(tan(x)-I)*sinh(1)}$  ${\displaystyle +Ci(tan(x)-I)*cosh(1))}$  ${\displaystyle +(1/2*I)*(I*Si(tan(x)+I)*sinh(1)}$  ${\displaystyle +Ci(tan(x)+I)*cosh(1))}$

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))

## 重积分

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

${\displaystyle \int \int (x^{2}*cos(x^{2})dxdx}$

-(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)

${\displaystyle \int \int (ln(tan(x)*cos(y))dxdy}$

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