Robert M. Corless
Department of Applied Mathematics
University of Western Ontario
London, Canada

Copyright 2001 by Robert M. Corless
All rights reserved

Useful one-word commands 

Использование монокоманд

Manipulations from calculus 

Манипуляции из начал анализа

int

Команда int

> restart;
> int( sin(x), x );

-cos(x)
> int( sin(x), x ) + C;

-cos(x)+C
> int( ln(x)/(1+x), x );

dilog(1+x)+ln(x)*ln(1+x)
> Digits := 30;

Digits := 30
> Int( exp(x^2), x=0..1 ) = int( exp(x^2), x=0..1 );

Int(exp(x^2),x = 0 .. 1) = -1/2*I*erf(I)*sqrt(Pi)
> evalf( % );

1.46265174590718160880404858686 = 1.462651745907181...
> int( x^3*sin(m*x), x );

1/m^4*(-m^3*x^3*cos(m*x)+3*m^2*x^2*sin(m*x)-6*sin(m...
> collect( %, [cos,sin], expand );

(-1/m*x^3+6/m^3*x)*cos(m*x)+(3/m^2*x^2-6/m^4)*sin(m...
> restart;
> int( tanh(x^2), x );

x+int(-2*1/(exp(x^2)^2+1),x)

> forget(int);
> infolevel[int] := 5;

infolevel[int] := 5
> int( tanh(x^2), x );
int/indef1: first-stage indefinite integration
int/indef2: second-stage indefinite integration
int/trighexp: case of integrand containing exp and hyperbolic trigs
int/indef1: first-stage indefinite integration
int/indef2: second-stage indefinite integration
int/trighexp: case of integrand containing exp and hyperbolic trigs
int/rischnorm: enter Risch-Norman integrator
int/rischnorm: exit Risch-Norman integrator
int/risch: enter Risch integration
int/risch/algebraic1: RootOfs should be algebraic numbers and functions
int/risch: the field extensions are

[x, exp(x^2)]
int/risch: Introduce the namings:

{_th[1] = exp(x^2)}
int/risch/int: integrand is

(_th[1]^2-1)/(_th[1]^2+1)

int/risch/int: integrand expressed as

1-2/(_th[1]^2+1)
int/risch/ratpart: integrating

-2*1/(_th[1]^2+1)
int/risch/ratpart: Hermite reduction yields

Int(-2*1/(_th[1]^2+1),x)
int/risch/ratpart:
Rothstein's method - resultant is:

(1-2*z*x)^2
nonconstant coefficients: integral is not elementary
int/indef1: first-stage indefinite integration
int/indef1: first-stage indefinite integration
int/indef2: second-stage indefinite integration
int/exp: case of integrand containing exp
int/prpexp: case ratpoly*exp(arg)
int/risch/exppoly: integrating

1
int/risch/int: integrand is

1
int/ratpoly/horowitz: integrating

1
int/risch/ratpoly: result is

x
int/risch/exppoly: integral of the "constant term" is

x
int/risch: exit Risch integration
int/indef1: first-stage indefinite integration
int/indef1: first-stage indefinite integration
int/indef2: second-stage indefinite integration
int/exp: case of integrand containing exp
int/prpexp: case ratpoly*exp(arg)
int/rischnorm: enter Risch-Norman integrator
int/rischnorm: exit Risch-Norman integrator

x+int(-2*1/(exp(x^2)^2+1),x)
> restart; 
> f := sin(3*arcsin(x));

f := sin(3*arcsin(x))
> int( f, x );

int(sin(3*arcsin(x)),x)
> F := -1/2*(2*x^2-3)*x^2;

F := -1/2*(2*x^2-3)*x^2
> restart;
> int( t^n, t=0..1 );

limit(-(t^(n+1)-1)/(n+1),t = 0,right)
> int( 1/(1+x^n), x=0..1 ); 
Definite integration: Can't determine if the integral is convergent.
Need to know the sign of --> n
Will now try indefinite integration and then take limits.

int(1/(1+x^n),x = 0 .. 1)

> assume( n > 0 );
> int( 1/(1+x^n), x=0..1 );

int(1/(1+x^n),x = 0 .. 1)
> int( 1/(a^8+x^8), x=0..1 );

1/8/a^8*LerchPhi(-1/(a^8),1,1/8)
> int( 1/(a^n+t^n), t=0..1 );

1/(a^n)*hypergeom([1, 1/n],[(n+1)/n],-a^(-n))
> restart;
> int( 1/sin(x), x=-1..1 );

undefined
> int( 1/sin(x), x=-1..1, CauchyPrincipalValue );

0
> int( 1/x^2, x=-1..1 );

infinity
> int( 1/x^3, x=-1..1 );

undefined
> restart;
> plots[setoptions](colour=BLACK);
> f := 1/(2+sin(x));

f := 1/(2+sin(x))
> plot( f, x=-3*Pi..3*Pi );

> F := int( f, x );

F := 2/3*sqrt(3)*arctan(1/3*(2*tan(1/2*x)+1)*sqrt(3...
> plot( F, x=-3*Pi..3*Pi, discont=true, colour=black );

> limit( F, x=Pi, right );

-1/3*Pi*sqrt(3)
> limit( F, x=Pi, left );

1/3*Pi*sqrt(3)
> int( f, x=-2*Pi..2*Pi );

4/3*Pi*sqrt(3)

> evalf( % );

7.255197458
> evalf( Int( f, x=-2*Pi..2*Pi ) );

7.255197457

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