An Example: Defining Your Own Integration Function
Now that we have introduced the basic features of patterns in
Mathematica, 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
Mathematica.
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
Mathematica.
| mathematical form | Mathematica definition |
 | integrate[y_+z_,x_]:=integrate[y,x]+integrate[z,x] |
 ( independent of  ) | integrate[c_y_,x_]:=c integrate[y,x]/;FreeQ[c,x] |
 | integrate[c_,x_]:=cx/;FreeQ[c,x] |
 ,  | integrate[x_^n_.,x_]:=x^(n+1)/(n+1)/;FreeQ[n,x]&&n!=-1 |
 | integrate[1/(a_.x_+b_.),x_]:=Log[ax+b]/a/;FreeQ[{a,b},x] |
 | integrate[Exp[a_.x_+b_.],x_]:=Exp[ax+b]/a/;FreeQ[{a,b},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.
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This makes
pull out factors that are independent of the integration variable
.
Mathematica tests each term in each product to see whether it satisfies the
FreeQ condition, and so can be pulled out.
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This gives the integral
of a constant.
Now the constant term in the sum can be integrated.
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This gives the standard formula for the integral of
. By using the pattern
, rather than
, we include the case of
.
Now this integral can be done completely.
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Of course, the built-in integration function
Integrate (with a capital
I) could have done the integral anyway.
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Here is the rule for integrating the reciprocal of a linear function. The pattern
stands for any linear function of
.
Here both
and
take on their default values.
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Here is a more complicated case. The symbol
now matches
.
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You can go on and add many more rules for integration. Here is a rule for integrating exponentials.
