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Manual Maple 2.3.2.mws

Comandos Úteis e Simples

Seção 2.3: Aplicações de Cálculo

Session 2.3.2: int

> restart;

> int( sin(x), x );

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

> int( ln(x)/(1+x), x );

> 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( % );

> 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*x)+6*m*x*cos(m*x))

> 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*x)

> restart;

> int( tanh(x^2), x );

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

> forget(int);

> 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

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:

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

int/risch/int:   integrand is

int/ratpoly/horowitz:   integrating

int/risch/ratpoly:   result is

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

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));

> int( f, x );

> 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 );

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

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

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

>

> restart;

> plots[setoptions](colour=BLACK);

> f := 1/(2+sin(x));

> plot( f, x=-3*Pi..3*Pi );

[Maple Plot]

> F := int( f, x );

> plot( F, x=-3*Pi..3*Pi, discont=true, colour=black );

[Maple Plot]

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

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

> int( f, x=-2*Pi..2*Pi );

> evalf( % );

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

>