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Practice Problems for Module 6

Sections 7.4 and 7.8

Directions. The following are review problems. We recommend you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it. You can also follow the link for the full video page to find related videos. 
  1. Evaluate the following integrals.
    1. \(\displaystyle \int \! \frac{3x^2+4x+3}{x^2+1} \, dx\)  

      \(\displaystyle \int \! \frac{3x^2+4x+3}{x^2+1} \, dx = 3x+2\ln|x^2+1|+C\)

      To see the full video page and find related videos, click the following link.
      WIR 20B M152 V34


    2. \(\displaystyle \int \! \frac{x^2-4x+2}{(x+2)^2(x+1)} \, dx\)  

      \(\displaystyle \int \! \frac{x^2-4x+2}{(x+2)^2(x+1)} \, dx = -6\ln|x+2|+\frac{14}{x+2}+7\ln|x+1|+C\)

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      WIR 20B M152 V35


    3. \(\displaystyle \int \! \frac{2x^3-4x^2+18x-11}{\left(x^2+9\right)\left(x^2+4\right)} \, dx\)  

      \(\displaystyle \int \! \frac{2x^3-4x^2+18x-11}{\left(x^2+9\right)\left(x^2+4\right)} \, dx = -\frac{5}{3}\arctan(3x)+\ln\left|x^2+4\right|+\frac{1}{2}\arctan(2x)+C\)

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      WIR 20B M152 V35.5


  2. Determine if the following integrals converge or diverge. If it converges, compute the value.
    1. \(\displaystyle \int_{-1}^2 \! \frac{9}{(x+1)^2} \, dx\)  

      Diverges

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      WIR 20B M152 V36


    2. \(\displaystyle \int_0^\infty \! \frac{5x^2}{\sqrt[3]{x^3+1}} \, dx\)  

      Diverges

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      WIR 20B M152 V37


    3. \(\displaystyle \int_1^5 \! (x-1)\ln(x-1) \, dx\)  

      Converges 
      \(\displaystyle \int_1^5 \! (x-1)\ln(x-1) \, dx = 8\ln(4)-4\)

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      WIR 20B M152 V38


    4. \(\displaystyle \int_3^\infty \! \frac{4}{3x^2+4x} \, dx\)  

      Converges 
      \(\displaystyle \int_3^\infty \! \frac{4}{3x^2+4x} \, dx=\ln\left(\frac{13}{9}\right)\)

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      WIR 20B M152 V39


  3. Use the Comparison Test to determine if the following integrals converge or diverge.
    1. \(\displaystyle \int_1^\infty \! \frac{3\sin^2(x)+2}{\sqrt{x}} \, dx\)  

      Diverges

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      WIR 20B M152 V40


    2. \(\displaystyle \int_2^\infty \! \frac{\cos(x)+5}{\sqrt{x^3+7}} \, dx\)  

      Converges

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      WIR 20B M152 V41


    3. \(\displaystyle \int_3^\infty \! \frac{10}{\sqrt{x}+5x\sqrt{x}} \, dx\)  

      Converges

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      WIR 20B M152 V42


    4. \(\displaystyle \int_1^\infty \! \frac{2+e^{-x}}{x} \, dx\)  

      Diverges

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      WIR 20B M152 V43