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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]


  • Differentiating the symbolic form gives the integrand back again.
  • In[2]:= D[%, x]


  • Here is a definite integral which cannot be done explicitly.
  • In[3]:= Integrate[f[x], {x, a[x], b[x]}]


  • This gives the derivative of the definite integral.
  • In[4]:= D[%, x]


  • Here is a definite integral with end points that do not explicitly depend onĀ x.
  • In[5]:= defint = Integrate[f[x], {x, a, b}]


  • The partial derivative of this with respect to u is zero.
  • In[6]:= D[defint, u]


  • There is a non-trivial total derivative, however.
  • In[7]:= Dt[defint, u]