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minor cleanup
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grzegorzmazur committed Feb 21, 2018
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Integration
===========

Integration can be performed by the function {Integrate}, which has
two calling conventions:
Integration can be performed by the function :func:`Integrate`, which has
two calling conventions::

* Integrate(variable) expression
* Integrate(variable, from, to) expression
Integrate(variable) expression
Integrate(variable, from, to) expression

Integrate can have its own set of rules for specific integrals, which
might return a correct answer immediately. Alternatively, it calls the
function AntiDeriv, to see if the anti-derivative can be determined
for the integral requested. If this is the case, the anti-derivative
is used to compose the output.
:func:`Integrate` can have its own set of rules for specific integrals, which
might return a correct answer immediately. Alternatively, it calls the function
:func:`AntiDeriv`, to see if the anti-derivative can be determined for the
integral requested. If this is the case, the anti-derivative is used to compose
the output.

If the integration algorithm cannot perform the integral, the
expression is returned unsimplified.

The integration algorithm
-------------------------

This section describes the steps taken in doing integration.

General structure
^^^^^^^^^^^^^^^^^

The integration starts at the function {Integrate}, but the task is
delegated to other functions, one after the other. Each function can
deem the integral unsolvable, and thus return the integral
unevaluated. These different functions offer hooks for adding new
types of integrals to be handled.
The integration starts at the function :func:`Integrate`, but the task is
delegated to other functions, one after the other. Each function can deem the
integral unsolvable, and thus return the integral unevaluated. These different
functions offer hooks for adding new types of integrals to be handled.

Expression clean-up
^^^^^^^^^^^^^^^^^^^

Integration starts by first cleaning up the expression, by calling
{TrigSimpCombine} to simplify expressions containing multiplications
of trigonometric functions into additions of trigonometric functions
(for which the integration rules are trivial), and then passing the
result to {Simplify}.
:func:`TrigSimpCombine` to simplify expressions containing multiplications of
trigonometric functions into additions of trigonometric functions (for which the
integration rules are trivial), and then passing the result to :func:`Simplify`.

Generalized integration rules
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Integration tables
^^^^^^^^^^^^^^^^^^

For elementary functions, Yacas uses integration tables. For instance,
For elementary functions, yacas uses integration tables. For instance,
the fact that the anti-derivative of :math:`\cos(x)` is :math:`\sin(x)` is
declared in an integration table.

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