Exercises for Section 6.1.  Complex Integrals

Formulas.   Let    where u(t) and v(t) are real-valued functions of the real variable t for   .  Then

(6-1)            .  

    We generally evaluate integrals of this type by finding the antiderivatives of u(t) and v(t) and evaluating the definite integrals on the right side of Equation (6-1).  

That is, if    and  ,  we have  

(6-2)            .  

Exercise 1.  Use Equations (6-1) and (6-2) to find  

1 (a).  .  

1 (b).  .  

1 (c).  .  

1 (d).  .

1 (e).  .

Exercise 2.  Let    be integers.  Show that  .  

Exercise 3.  Show that    provided  .  

Exercise 4.  Given that    and    are continuous on  .  Establish the following:

4 (a).  Identity (6-3)   .  

4 (b).  Identity (6-4)   .  

4 (c).  Identity (6-6)   .  

4 (d).  Identity (6-7)   .   

Exercise 5.  Let  ,  where  u  and  v  are differentiable.   Show that

        .  

Exercise 6.  Use integration by parts to verify that  .

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