Solution 5.

See text and/or instructor's solution manual.

Answer.   Apply Leibniz's rule

        
        
Hence the integrand vanishes and    which implies that    is harmonic.  

Solution.   Calculate the partial derivatives of the term     in the integrand  

                         and     .  

Then calculate the second partial derivatives  

                         and     
                    
                    

Now apply Leibniz's rule

        

Hence   satisfies Laplace's equation   .

Therefore,    is a harmonic function.  

We are done.   

Aside.  We can let Mathematica double check our work.

Now add them together.

Hence the integrand vanishes which implies that   .  

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell