Math231       Exam 1                                            Spring, 2018

 

Multiple choice (6 points each)                    (Five possible answers provided)

 

 

                      π/2

1.  Evaluate   ∫ sin6x cos3x dx                               Answer:  2/63

                     0

 

2.  Use a trig substitution to transform integral   ∫ [ x2 / (√(4 + x2) ] dx

 

                                              Answer:   ∫ 4 tan2θ secθ  dθ


                      4

3.  Evaluate   ∫ x lnx  dx        Answer:  8 ln|4| - 15/4

                     1

 

4.  Find the correct partial fraction decomposition of  10x2 / (x-1)2(x2+1)

 

                                 Answer:  A/(x-1)  +  B/(x-1)2  +  (Cx + D)/(x2 + 1)


                     1

5.  Evaluate  ∫ arctan(2x) dx     Answer:  arctan(2) - (1/4) ln|5|

                    0


                     5

6.  Evaluate  ∫ dx / (x2 - 4x +13)         Answer:  (1/3) arctan (1)

                    2

 

7.  Use a substitution to transform the integral  ∫ sec5x tan3x dx  into an integral

     involving a new variable, u.              Answer:  ∫ (u6 - u4) du

 

 

Determine if the improper integrals 8 through 12 converge or diverge.   (3 points each)


     ∞

8.  ∫ (1/xp) dx where  p > 1                                           Answer:  Converges

    1

 

     ∞

9.  ∫ [{ √(9+x2) + sin2x }/ {3x2} ] dx                           Answer:  Diverges

    1

 

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