WOLFRAM LANGUAGE TUTORIAL
An Example: Defining Your Own Integration Function
Now that we have introduced the basic features of patterns in the Wolfram Language, we can use them to give a more or less complete example. We will show how you could define your own simple integration function in the Wolfram Language.
From a mathematical point of view, the integration function is defined by a sequence of mathematical relations. By setting up transformation rules for patterns, you can implement these mathematical relations quite directly in the Wolfram Language.
|mathematical form||Wolfram Language definition|
| ( independent of )||integrate[c_y_,x_]:=c integrate[y,x]/;FreeQ[c,x]|
Definitions for an integration function.
This implements the linearity relation for integrals:
The associativity of Plus
makes the linearity relation work with any number of terms in the sum.
This makes integrate
pull out factors that are independent of the integration variable x
The Wolfram Language tests each term in each product to see whether it satisfies the FreeQ
condition, and so can be pulled out.
This gives the integral
of a constant.
Now the constant term in the sum can be integrated.
This gives the standard formula for the integral of
. By using the pattern x_^n_.
, rather than x_^n_
, we include the case of
Now this integral can be done completely.
Of course, the built‐
in integration function Integrate
(with a capital I
) could have done the integral anyway.
Here is the rule for integrating the reciprocal of a linear function. The pattern a_.x_+b_.
stands for any linear function of x
Here both a
take on their default values.
Here is a more complicated case. The symbol a
now matches 2p
You can go on and add many more rules for integration. Here is a rule for integrating exponentials.