This is documentation for Mathematica 4, which was
based on an earlier version of the Wolfram Language.
Wolfram Research, Inc.

3.5.9 Manipulating Integrals in Symbolic Form

When Mathematica cannot give you an explicit result for an integral, it leaves the integral in a symbolic form. It is often useful to manipulate this symbolic form.

Mathematica cannot give an explicit result for this integral, so it leaves the integral in symbolic form.

In[1]:= Integrate[x^2 f[x], x]

Out[1]=

Differentiating the symbolic form gives the integrand back again.

In[2]:= D[%, x]

Out[2]=

Here is a definite integral which cannot be done explicitly.

In[3]:= Integrate[f[x], {x, a[x], b[x]}]

Out[3]=

This gives the derivative of the definite integral.

In[4]:= D[%, x]

Out[4]=

Here is a definite integral with end points that do not explicitly depend on x.

In[5]:= defint = Integrate[f[x], {x, a, b}]

Out[5]=

The partial derivative of this with respect to u is zero.

In[6]:= D[defint, u]

Out[6]=

There is a non-trivial total derivative, however.

In[7]:= Dt[defint, u]

Out[7]=