Calculus 2  Final Exam    -   Math 230   Spring 2006

 

 

1.

 

 

Evaluate the following integral.    ∫ [x lnx] dx           Hint:  Try integration by parts.

 

Answer:  I  =  (x2 /2) ln x – ¼ x2  +  C

 

 

 

2.

 

Evaluate the integral         ∫ tan 5x dx          Hint:  A substitution will help.

 

 Answer:  I  =  -(1/5) ln (cos 5x)  + C

 

 

 

3.

 

Evaluate the integral

 

   ∫ [(2x2 + 14x + 49)/(x3 – 7x2)] dx        Hint:  Try partial fractions.

 

Answer:  I  =  - 3 ln x  +  7/x  - 5 ln (x – 7)  +  C

 

 

4.

 

 

 

Evaluate the integral      ∫ 1 / [√ (x2 + 25) ] dx          Hint:  Try a trig substitution.

 

Answer:  I  =  ln [ √({x2 + 25)  +  x}/5]   +  C

 

 

 

 

5.

 

 

Does the improper integral converge or diverge.  If it converges, find its value.

        ∞

        ∫  x exp(-x2)  dx

0                         

 

Hint:  Try a substitution to evaluate integral.   i.e.  u = -x2

 

Answer:    Converges.  Value = 1/2

 

 

 

 

 

 

 

 

6.

 

Find the third partial sum, S3, for the series:

 

          ∞                                           

a.       ∑ 1 / 2n             

        n=1   

 

 Answer:  7/8

 

b.  If we know that for ∑ an   the sequence of partial sums satisfies   lim  Sn =  4

                                                                                                           n à ∞

    can we conclude that  ∑ an   converges?  Explain your answer.

 

Answer:  If limit of Sn exists as n à ∞, then the partial sums approach the

                limiting value and the series converges.

 

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