7.8: Homework- Integration by Parts

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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
Watch the following Khan Academy video: Integration by parts twice
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
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
Watch the following Khan Academy video: Integration by parts with e and cos together.
Use integration by parts to find \(\int e^x \sin(x)dx\).
\(\frac{\sin(x) e^x - \cos(x) e^x}{2} + C\)
ans
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