Calculus 2  Hour Exam 1  (continued)  -   Math 231     Spring, 2012

 

 

 

 

 

 

 

7.

   

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For the integral  Evaluate  ∫ dx / (x - 3)1/4      do the following:

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a.        Explain why it is improper

b.      Write the integral as the limit of (proper) integrals

c.       Determine whether the integral converges or diverges. 

If it converges, find its value.

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Answers:  a. Integrand unbounded at x=3    b.  lim    ∫ dx / (x - 3)1/4      c. 4/3

                                                                          t→∞  t

 

 

 

 

8.

 

Use the comparison test to determine whether the improper integral converges

or diverges.        

                              ∞

                               ∫ ( e-x / √x ) dx                   Answer:   Converges

                              1

 

 

9.

 

 

Find    ∫ dx / (x3 + 4x)      Answer:  (1/4 lnx  -  1/8 ln | x2 + 4 |  + C

                                                             

 

10.

 

 

Find   ∫ √ ( 25 – x2 ) dx              Answer:  (25/2) sin-1(x/5) + (x/2) √(25 – x2 )  + C

 

11.

 

 

Find  ∫ x3 ln x dx              Answer:  (1/4) x4  lnx - (1/16) x4  + C