Calculus 2  Hour Exam 1   -   Math 231     Fall, 2010

 

 

 

 

 

1.

 

 

a. Calculate the integral     I  =   ∫ x e5x  dx   

 

b. Write out the form of the partial fraction decomposition for the function

 

                        (x4  +  1) / (x5  +  4x3)  

 

Answers:   a.   I  =   (1/5) x e5x  -  (1/25) e5x   +  C

 

                  b.   A/x  +  B/x2   +  C/x3  +  (Dx + E)/(x2 + 4)

 

 

 

 

 

 

2.

 

                                                             ∞

a.        Compute the integral       I  =  ∫ [1 / (3x + 1)2 ]dx     

                                                             1

b.      Use the comparison Theorem to determine if the following converges or diverges.

 

                                              ∞

                           I  =  ∫ [ ( x + 1 ) / √( x4 – x ) ]dx    

                                              1

 

                                                         Answers:      a. 1/12            b.  Integral diverges

 

 

 

 

3.

 

 

a.       Compute         ∫ 1 / √(x2 – 4) dx

 

b.      Compute         ∫  tan3 θ sec θ  dθ

 

Answers:   a.  ln | ½ [x  + √(x2 – 4)] |  +  C             b.  (1/3) sec3 θ  - sec θ  +  C

 

 

 

 

 

 

 

4.

 

                             

Do one of the following problems.

 

Compute     a.   ∫ 1 / (x2  -  2x  +  5) dx       or         

 

                    b.    ∫ x2 / (x2  -  6x  +  5) dx   

 

Answers:   a.  (1/2) tan-1 (x – 1)/2  +  C    

 

      b.   x  +  (25/4) ln | x – 5 |  - (1/4) ln | x – 1 |  +  C

 

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