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Home / Questions /Q 1350
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Tony Spark
Tony Spark

Tony Spark

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Asked: 2020-04-25T15:49:53+00:00 In:

Integration by parts integral udv = uv – integral vdu * u = x dv = e^2 times dx du = dx v = ze =x…

| 그_1___1ntegration by Tarts udVE uv-Tvdu dueaxーーー U2 ーーーーーー 24 2X -2-le e2

    1 Answer

    1. The formula you used and the procedure you followed is correct
      .
      i will do it once more
      int{x e^{2 x} d x}
      use integration by parts
      int operatorname{u} operatorname{dv} = operatorname{u}operatorname{v} - int operatorname{v} operatorname{du}
      Let operatorname{u}=x and operatorname{dv}=e^{2 x} dx
      Then operatorname{du}=left(xright)^{prime }dx=dx and operatorname{v}=int{e^{2 x} d x}=frac{e^{2 x}}{2}
      Integral becomes
      {int{x e^{2 x} d x}}={left(x cdot frac{e^{2 x}}{2}-int{frac{e^{2 x}}{2} cdot 1 d x}right)}
      ={left(frac{x}{2} e^{2 x} - int{frac{e^{2 x}}{2} d x}right)}
        = frac{xe^{2 x}}{2} - {left(frac{1}{2} int{e^{2 x} d x}right)}
        = frac{xe^{2 x}}{2} - frac{1}{2} int{e^{2 x} d x}
        = frac{xe^{2 x}}{2} - frac{1}{2} *frac{e^{2 x}}{2}
      = frac{xe^{2 x}}{2} - frac{e^{2 x}}{4}
      Simplify:

      int{x e^{2 x} d x} = left(frac{x}{2} - frac{1}{4}right) e^{2 x}

      Add the constant of integration:
      int{x e^{2 x} d x} = left(frac{x}{2} - frac{1}{4}right) e^{2 x}+C

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