Calculus 2  Final Exam    -   Math 231   Spring 2007

 

 

 

1.

 

 

Use a trigonometric substitutions to evaluate the integral

                                                                                         

     ∫ [x2/(x2 + 1)7/2] dx            Hint:  Use  x  =  tan θ

 

Answer:  I  =  (1/3)x3/(x2 + 1)3/2  -  (1/5)x5/(x2 + 1)5/2 +  C

 

 

 

 

 

2.

 Find the integrals

 

a.     ∫ x3 ln x dx             b.       ∫ dx /[x(x + 1)]

 

Hint for a:  Use integration by parts.   Answer:  I  =  (x4/4)ln|x|  - (1/16)x4  +  C

 

Hint for b:  Use partial fractions    Answer:  I  =  ln|x|  - ln|x + 1|  +  C

 

 

 

 

 

3.

 

 

 Evaluate each improper integral, or determine that it diverges (use the result of problem 2b).

        1                                         ∞                                        ∞

a.       ∫ dx /[x(x + 1)]             b.  ∫ dx /[x(x + 1)]              c.  ∫ dx /[x(x + 1)]

0                                       1                                         0

 

Hint:  Use result of part b in Problem 2.

 

Answer for a:  diverges     Answer for b:  converges      Answer for c:  diverges

 

 

 

 

 

 

 

4.

 

 Classify the series as absolutely convergent, conditionally convergent or divergent.  Say

what tests you use.

      ∞                                           ∞

 a.  ∑ n! / 2007n                     b.  ∑ (-1)k / (k2/3 + ln k)

    n=0                                        k=1

 

Hint for a:  Use ratio test.     Answer:  diverges

 

Hint for b:  Use alternating series test.  Then use limit comparison test.

 

Answer:  conditionally convergent

 

 

 

 

 

5.

                                                                                  ∞

Find the interval of convergence of the power series    ∑ xn/nen

                                                                                                  n=1

 

Hint:  radius of convergence is  e.

 

Answer:  [ -e, e )

 

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