1. (17 Points) Use Integration by Parts to Find

Math 116 Exam 2 Fall 2013

Name:______This is a closed book exam. You may use a calculator and the formulas handed out with the exam. You may find that your calculator can do some of the problems. If this is so, you still need to show how to do the problem by hand. In other words, show all work and explain any reasoning which is not clear from the computations. (This is particularly important if I am to be able to give part credit.) Turn in this exam along with your answers. However, don't write your answers on the exam itself; leave them on the pages with your work. Also turn in the formulas; put them on the formula pile.

1. (17 points) Use integration by parts to find

2. (17 points) Find (use a trig identity for cos2x).

3. (18 points) Find . (use a trig substitution)

4. (16 points) Complete the square and make a substitution that transforms the integral into one of the following forms: or or where a is a positive number you are to find. You don't have to evaluate the resulting integral that you obtain after completing the square and making the substitution.

5. (17 points) Use the partial fraction method to express as the sum of simpler fractions.

6. (15 points) Use the midpoint method with n = 6 subintervals to approximate

Solutions

1. Let u = x and dv = cosh(2x) dx. So du = dx and v = = sinh(2x). } 5 points

Therefore = - . } 6 points

Therefore = - cosh(2x). } 6 points

2. = } 4 points

Therefore = = } 4 points

Therefore = = } 9 points

3. Let x = 4 sin u. So dx = 4 cos u du. Thus = = 16 } 8 points

Thus = 16 = 8( u - ) } 7 points

3 points

10 points

Let u = x – 2. Then du = dx and = } 6 points

5. One has = + + where A, B and C are to be determined. } 6 points

Multiply by (x2)2(x+1) to get 1=A(x+1) + B(x – 2)(x+1) + C(x2)2. } 2 points

9 points

6. The midpoint method says » 2(Dx)[ f(x1) + f(x3) + … + f(xn-1) ] where Dx = and xj = a + j(Dx). In the case of with n = 6 one has Dx = 1 and xj = j. So » = = = = = 19.34