Calculus 2  Hour Exam 1   -   Math 231     Spring, 2012

 

1.

 

Using a trigonometric substitution, transform the integral  ∫ dx / (x2 + 1)5/2 

 

Answer:    ∫ cos3θ dθ

 

 

2.

 

 

                                                                                ∞

Determine all values of  p  for which the integral  ∫ dx / xp  converges.

                                                                                0

Answer:  p > 1

 

 

 

 

3.

 

 

Write out the general form of the partial fraction decomposition of the function

 

            ( 8 x3 – 7 x2 + x + 3 ) / [ (x2 + 2) (x2 – 1) ]

 

Answer:   (Ax + B) / (x2 + 2) +  C / (x -1) + D / (x + 1)

 

 

4.

 

 

Evaluate  ∫ x e-4x dx              Answer:  -(1/4) x e-4x  - (1/16) e-4x  + C

 

5.

 

 

Evaluate  ∫ cos3x sin2x dx              Answer:  (1/3) sin3x - (1/5) sin5x  + C

 

 

6.

 

                ∞

Evaluate  ∫ dx / (x2 + 1)              Answer:  π / 4

                1

 

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