Manipulating Integrals in Symbolic Form

When the Wolfram Language 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.

The Wolfram Language cannot give an explicit result for this integral, so it leaves the integral in symbolic form:

Integrate[x^2 f[x],x]

Differentiating the symbolic form gives the integrand back again:

D[%,x]

Here is a definite integral which cannot be done explicitly:

Integrate[f[x],{x,a[x],b[x]}]

This gives the derivative of the definite integral:

D[%,x]

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

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

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

D[defint,u]

There is a non-trivial total derivative, however:

Dt[defint,u]