Homework:  Integration by Parts

71 Homework: Integration by Parts

Solve each of the following using integration by parts:
1. [latex]\int x \cos(x)dx[/latex]
  
  [latex]x \sin(x) + \cos(x) + C[/latex]
  
  ans
2. [latex]\int (4x - 1) \cos(x)dx[/latex]
  
  [latex](4 x - 1) \sin(x) + 4 \cos(x) + C[/latex]
  
  ans
3. [latex]\int x \sin(x)dx[/latex].
  
  [latex]-x \cos(x) + \sin(x) + C[/latex]
  
  ans
4. [latex]\int x e^xdx[/latex].
  
  [latex]x e^x - e^x + C[/latex]
  
  ans
5. [latex]\int \ln(x)dx[/latex]. (Hint: Let [latex]u = \ln(x)[/latex] and [latex]v' = 1[/latex])
  
  [latex]x \ln(x) - x[/latex]
  
  ans
Watch the following Khan Academy video: Integration by parts twice
Use integration by parts to solve [latex]\int x^2 \cos(x)dx[/latex].

[latex]x^2 \sin(x) + 2 x \cos(x) - 2\sin(x) + C[/latex]

ans
Use integration by parts to solve [latex]\int x^3 e^xdx[/latex].

[latex]x^3 e^x - 3x^2 e^x + 6x e^x - 6 e^x + C[/latex]

ans
Watch the following Khan Academy video: Integration by parts with e and cos together.
Use integration by parts to find [latex]\int e^x \sin(x)dx[/latex].

[latex]\frac{\sin(x) e^x - \cos(x) e^x}{2} + C[/latex]

ans
Two part question:
1. Use [latex]u[/latex]-substitution to find [latex]\int \sin(2x)dx[/latex] and [latex]\int \cos(2x)dx[/latex].
  
  [latex]-\frac{1}{2} \cos(2x)[/latex] and [latex]\frac{1}{2} \sin(2x)[/latex]
  
  ans
2. Use integration by parts to find [latex]\int x \sin(2x)dx[/latex].
  
  [latex]-\frac{1}{2} x \cos(2x) - \frac{1}{4} \sin(2x)[/latex]
  
  ans

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