Example 8.19.  Evaluate  .  

Explore Solution 8.19.

We must guide Mathematica and write an appropriate form of the anti-derivative.

The integrand is discontinuous at x = 0 where it has a singularity.  Indeed, Mathematica has a branch cut for    along the negative x-axis and maps these points onto the ray .  Hence the function    does not have the graph as taught in elementary calculus.  We must use the following function  h[x].

Now we pursue the Cauchy principal value of the integral.

Warning. The following computations will not produce the principal value of the real integral.  Indeed, Mathematica has a branch cut for    along the negative x-axis and maps these points onto the ray ,  in particular .  

Correction. The following choice for the form for the indefinite integral will produce the desired computation.

Which is also the correct answer.

Remark. The PrincipalValue option does not give and answer to this problem.

We are done.

Aside.  Observe that  

(c) 2006 John H. Mathews, Russell W. Howell