3.5.6 Indefinite Integrals

The Mathematica function Integrate[f, x] gives you the indefinite integral . You can think of the operation of indefinite integration as being 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 which 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, or indeed by adding any function that is constant except at discrete points.
If you fill in explicit limits for your integral, any such constants of integration must cancel out. But even though the indefinite integral can have arbitrary constants added, it is still often very convenient to manipulate it without filling in the limits.

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.

Another assumption that Integrate implicitly makes is that all the symbolic quantities in your integrand have "generic" values. Thus, for example, Mathematica will tell you that is even though this is not true in the special case .

You should realize that the result for any particular integral can often be written in many different forms. Mathematica tries to give you the most convenient form, following principles such as avoiding explicit complex numbers unless your input already contains them.