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Definite IntegralsDifferential Equations

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]=

Definite IntegralsDifferential Equations