Integration

Some of the following procedures involve the use of the derivatives and you should be
careful when the interval evaluation of these derivative cannot be
performed. For example when considering
you should
not 0 in the integration domain. In that case you should integrate
around 0 by using the `Integrate` problem while using another
procedure for the remaining part of the domain.

The basic integration procedure is `Integrate(Func , Init)` where
`Func` is the integrand and `Init` the range for the
integration. For example

Integrate(1/x,[1,2]);returns the range [.69264842756679057, .69364643155883843] if

For accurate computation for function that are at least twice
differentiable you may use `IntegrateTrapeze` or `IntegrateRectangle` which have the same syntax than `Integrate`. You may also use `IntegrateTaylor` that use a Taylor
expansion of order `N` of the function. The syntax is

IntegrateTaylor(Func,Init,N)

All these procedures will return `UNABLE` if the integral cannot
be computed with the desired accuracy or `FAILED` if the number of
boxes exceed the global variable
``ALIAS/maxbox``. As the default value of this variable is 20 000
(i.e. relatively small) it may be necessary to set the variable to a
larger number before calculating the integral.

Note that the derivatives of the function and the function itself may
involve expression that cannot be interval evaluated everywhere.
Hence it is of good policy to check this point by using the
`Problem_Expression` package and that may require to set some
parameters of this package (see section 2.1.5).

For example assume that we have to integrate the function
in the range 0, . The derivatives of this
expression involves the expression of
which
is not defined in 0. If we use `IntegrateTaylor` of
order 2 we need to ensure that the function and the derivatives up to
the third one can be interval evaluated.

eq:=sqrt(1-cos(x)^2): `ALIAS/close_to_zero`:=1e-20: `ALIAS/low_value_expr_violated`:=1: `ALIAS/high_value_expr_violated`:=1: Verify_Problem_Expression([eq,diff(eq,x\$1),diff(eq,x\$2),diff(eq,x\$3)],[x]): IntegrateTaylor(eq,[0,evalf(Pi)/2.],2);The setting of the parameters allows to calculate the integral.

The basic multiple integral procedure of `ALIAS` is

IntegrateMultiRectangle(Func,Vars,Init)that may be used as in this example:

IntegrateMultiRectangle(x*sin(y),[x,y],[[0,1],[0,1]]);This procedure requires that the function is at least twice differentiable.

A more sophisticated procedure is based on Taylor expansion. Its syntax is

IntegrateMultiTaylor(Func,Vars,Init,N,Type)where

2018-07-25