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7.8: Homework- Integration by Parts

  • Page ID
    88691
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    1. Solve each of the following using integration by parts:
      1. \(\int x \cos(x)dx\)
        \(x \sin(x) + \cos(x) + C\)
        ans
      2. \(\int (4x - 1) \cos(x)dx\)
        \((4 x - 1) \sin(x) + 4 \cos(x) + C\)
        ans
      3. \(\int x \sin(x)dx\).
        \(-x \cos(x) + \sin(x) + C\)
        ans
      4. \(\int x e^xdx\).
        \(x e^x - e^x + C\)
        ans
      5. \(\int \ln(x)dx\). (Hint: Let \(u = \ln(x)\) and \(v' = 1\))
        \(x \ln(x) - x\)
        ans
    2. Watch the following Khan Academy video: Integration by parts twice
    3. Use integration by parts to solve \(\int x^2 \cos(x)dx\).
      \(x^2 \sin(x) + 2 x \cos(x) - 2\sin(x) + C\)
      ans
    4. Use integration by parts to solve \(\int x^3 e^xdx\).
      \(x^3 e^x - 3x^2 e^x + 6x e^x - 6 e^x + C\)
      ans
    5. Watch the following Khan Academy video: Integration by parts with e and cos together.
    6. Use integration by parts to find \(\int e^x \sin(x)dx\).
      \(\frac{\sin(x) e^x - \cos(x) e^x}{2} + C\)
      ans
    7. Two part question:
      1. Use \(u\)-substitution to find \(\int \sin(2x)dx\) and \(\int \cos(2x)dx\).
        \(-\frac{1}{2} \cos(2x)\) and \(\frac{1}{2} \sin(2x)\)
        ans
      2. Use integration by parts to find \(\int x \sin(2x)dx\).
        \(-\frac{1}{2} x \cos(2x) - \frac{1}{4} \sin(2x)\)
        ans

    This page titled 7.8: Homework- Integration by Parts is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Tyler Seacrest via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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