15.1 Indefinite Integrals
Indefinite integration.
The Mathematica TE function Integrate[f, x] gives you the indefinite integral 
. You can think of the process of indefinite integration as like an inverse of differentiation. If you take the result from Integrate[f, x], and then differentiate it, you always get a result that is mathematically equal to the original expression f.
In general, however, there is a whole family of results that have the property that their derivative is f. Integrate[f, x] gives you an expression whose derivative is f. You can get other expressions by adding an arbitrary constant of integration.
When you fill in explicit limits for your integral, any such constants of integration must cancel out. Even though the indefinite integral contains arbitrary constants, it is still often very convenient to manipulate it before filling in the limits.
Mathematica TE applies standard rules to find indefinite integrals.
In[1]:= Integrate[x^2, x]
Out[1]= 
You can add an arbitrary constant to the indefinite integral, and still get the same derivative. Integrate simply gives you an expression with the required derivative.
In[2]:= D[ % + c , x]
Out[2]= 
This gives the indefinite integral 
.
In[3]:= Integrate[1/(x^2 - 1), x]
Out[3]= 
Differentiating should give the original function back again.
In[4]:= D[%, x]
Out[4]= 
You need to manipulate it to get it back into the original form.
In[5]:= Simplify[%]
Out[5]= 
The Integrate function assumes that any object that does not explicitly contain the integration variable is independent of it, and can be treated as a constant. As a result, Integrate is like an inverse of the partial differentiation function D.
The variable a is assumed to be independent of x.
In[6]:= Integrate[a x^2, x]
Out[6]= 
The integration variable can be any expression.
In[7]:= Integrate[x b[x]^2, b[x]]
Out[7]= 
Another assumption that Integrate implicitly makes is that all the symbolic quantities in your integrand have "generic" values. Thus, for example, Mathematica TE will tell you that 
is 
even though this is not true in the special case 
.
Mathematica TE gives the standard result for this integral, implicitly assuming that n is not equal to -1.
In[8]:= Integrate[x^n, x]
Out[8]= 
If you specifically give an exponent of -1, Mathematica TE produces a different result.
In[9]:= Integrate[x^-1, x]
Out[9]= 
