2.3.14 An Example: Defining Your Own Integration FunctionNow 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] | | | integrate[c_ y_, x_] :=
c integrate[y, x] /; FreeQ[c, x] |  | integrate[c_, x_] := c x /; FreeQ[c, x] |  ,  | integrate[x_^n_., x_] := x^(n+1)/(n+1) /;
FreeQ[n, x] && n  -1 |  | integrate[1/(a_. x_ + b_.), x_] :=
Log[a x + b]/a /; FreeQ[{a,b}, x] |  | integrate[Exp[a_. x_ + b_.], x_] :=
Exp[a x + b]/a /; FreeQ[{a,b}, x] |
Definitions for an integration function. This implements the linearity relation for integrals:  . | |
In[1]:=
integrate[y_ + z_, x_] :=
integrate[y, x] + integrate[z, x]
|
|
| The associativity of Plus makes the linearity relation work with any number of terms in the sum. | |
In[2]:=
integrate[a x + b x^2 + 3, x]
|
Out[2]=
|
|
| This makes integrate pull out factors that are independent of the integration variable x. | |
In[3]:=
integrate[c_ y_, x_] := c integrate[y, x] /; FreeQ[c, x]
|
|
| Mathematica tests each term in each product to see whether it satisfies the FreeQ condition, and so can be pulled out. | |
In[4]:=
integrate[a x + b x^2 + 3, x]
|
Out[4]=
|
|
This gives the integral  of a constant. | |
In[5]:=
integrate[c_, x_] := c x /; FreeQ[c, x]
|
|
| Now the constant term in the sum can be integrated. | |
In[6]:=
integrate[a x + b x^2 + 3, x]
|
Out[6]=
|
|
| Now this integral can be done completely. | |
In[8]:=
integrate[a x + b x^2 + 3, x]
|
Out[8]=
|
|
| Of course, the built-in integration function Integrate (with a capital I) could have done the integral anyway. | |
In[9]:=
Integrate[a x + b x^2 + 3, x]
|
Out[9]=
|
|
| Here is the rule for integrating the reciprocal of a linear function. The pattern a_. x_ + b_. stands for any linear function of x. | |
In[10]:=
integrate[1/(a_. x_ + b_.), x_] :=
Log[a x + b]/a /; FreeQ[{a,b}, x]
|
|
| Here both a and b take on their default values. | |
In[11]:=
integrate[1/x, x]
|
Out[11]=
|
|
| Here is a more complicated case. The symbol a now matches 2 p. | |
In[12]:=
integrate[1/(2 p x - 1), x]
|
Out[12]=
|
|
| You can go on and add many more rules for integration. Here is a rule for integrating exponentials. | |
In[13]:=
integrate[Exp[a_. x_ + b_.], x_] :=
Exp[a x + b]/a /; FreeQ[{a,b}, x]
|
|
|
(
independent of 
) 
. By using the pattern 
. 
