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The Integral Practice Test

By — McGraw-Hill Professional
Updated on Sep 1, 2011

Review the following concepts if needed:

The Integral Practice Test

  1. Calculate each of the following antiderivatives:

    (a) Antiderivative of x 2 − sin x

    (b) Antiderivative of e 3 x + x 4 − 2

    (c) Antiderivative of 3 t 2 + (In t/t )

    (d) Antiderivative of tan x − cos x + sin 3 x

    (e) Antiderivative of cos 3 x + sin 4 x + 1

    (f) Antiderivative of (cos x ) · e sin x 

  2. Calculate each of the following indefinite integrals:

    (a) ∫ x sin x 2 dx

    (b) ∫ (3/ x ) In x 2 dx

    (c) ∫ sin x · cos x dx

    (d) ∫ tan x · In cos x dx

    (e) ∫ sec 2 x · e tan x dx

    (f) ∫ (2 x + 1) · ( x 2 + x + 7) 43 dx

  3. Use Riemann sums to calculate each of the following integrals:

    The Integral Exercises

  4. Use the Fundamental Theorem of Calculus to evaluate each of the following integrals:

    The Integral Exercises

  5. Calculate the area under the given function and above the x -axis over the indicated interval.

    (a) f ( x ) = x 2 + x + 6 [2, 5]

    (b) g ( x ) = sin x cos x [0, /4]

    (c) h ( x ) = xe x 2 [1, 2]

    (d) k ( x ) = In x/x [1, e ]

  6. Draw a careful sketch of each function on the given interval, indicating subintervals where area is positive and area is negative.

    (a) f ( x ) = x 3 + 3 x [−2, 2]

    (b) g ( x ) = sin 3 x cos 3 x [−2 , 2 ]

    (c) h ( x ) = In x/x [1/2, e ]

    (d) m ( x ) = x 3 e x 4 [−3, 3]

  7. For each function in Exercise 6, calculate the positive area between the graph of the given function and the x -axis over the indicated interval.

  8. In each part of Exercise 6, calculate the signed area between the graph of the given function and the x -axis over the indicated interval.

  9. Calculate the area between the two given curves over the indicated interval.

    The Integral Exercises

  10. Calculate the area enclosed by the two given curves.

    The Integral Exercises

Answers

  1. (a) F ( x ) = x 3 /3 + cos x + C

    (b) F ( x ) = e 3 x /3 + x 5 /5 − 2 x + C

    (c) F ( t ) = t 3 + [ln t ] 2 /2

    (d) F ( x ) = −ln(cos x ) − sin x − [cos 3 x ]/3 + C

    (e) F ( x ) = [sin 3 x ]/3 − [cos4 x ]/4 + x + C

    (f) F ( x ) = e sin x + C

  2. (a) SOLUTIONS TO EXERCISES Chapter 4 

    SOLUTIONS TO EXERCISES Chapter 4

  3. (a) We have

    SOLUTIONS TO EXERCISES Chapter 4

    (b) We have

    SOLUTIONS TO EXERCISES Chapter 4

  4.  

    SOLUTIONS TO EXERCISES Chapter 4

    SOLUTIONS TO EXERCISES Chapter 4

     

    SOLUTIONS TO EXERCISES Chapter 4

  5.  

    SOLUTIONS TO EXERCISES Chapter 4

  6.  

    SOLUTIONS TO EXERCISES Chapter 4

    Fig. S4.6(a)

    SOLUTIONS TO EXERCISES Chapter 4

    Fig. S4.6(b)

     

    SOLUTIONS TO EXERCISES Chapter 4

    Fig. S4.6(c)

    SOLUTIONS TO EXERCISES Chapter 4

    Fig. S4.6(d)

  7.  

    SOLUTIONS TO EXERCISES Chapter 4

    SOLUTIONS TO EXERCISES Chapter 4

    SOLUTIONS TO EXERCISES Chapter 4

     

    SOLUTIONS TO EXERCISES Chapter 4

  8.  

    SOLUTIONS TO EXERCISES Chapter 4

     

    SOLUTIONS TO EXERCISES Chapter 4

  9.  

    SOLUTIONS TO EXERCISES Chapter 4

    SOLUTIONS TO EXERCISES Chapter 4

  10.  

    SOLUTIONS TO EXERCISES Chapter 4

     

    SOLUTIONS TO EXERCISES Chapter 4

    SOLUTIONS TO EXERCISES Chapter 4

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