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-4*x*exp(-2*x)

Integral of -4*x*exp(-2*x) dx

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The solution

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  1              
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 |        -2*x   
 |  -4*x*e     dx
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014xe2xdx\int\limits_{0}^{1} - 4 x e^{- 2 x}\, dx
Integral((-4*x)*exp(-2*x), (x, 0, 1))
Detail solution
  1. Use integration by parts:

    udv=uvvdu\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

    Let u(x)=4xu{\left(x \right)} = - 4 x and let dv(x)=e2x\operatorname{dv}{\left(x \right)} = e^{- 2 x}.

    Then du(x)=4\operatorname{du}{\left(x \right)} = -4.

    To find v(x)v{\left(x \right)}:

    1. Let u=2xu = - 2 x.

      Then let du=2dxdu = - 2 dx and substitute du2- \frac{du}{2}:

      (eu2)du\int \left(- \frac{e^{u}}{2}\right)\, du

      1. The integral of a constant times a function is the constant times the integral of the function:

        False\text{False}

        1. The integral of the exponential function is itself.

          eudu=eu\int e^{u}\, du = e^{u}

        So, the result is: eu2- \frac{e^{u}}{2}

      Now substitute uu back in:

      e2x2- \frac{e^{- 2 x}}{2}

    Now evaluate the sub-integral.

  2. The integral of a constant times a function is the constant times the integral of the function:

    2e2xdx=2e2xdx\int 2 e^{- 2 x}\, dx = 2 \int e^{- 2 x}\, dx

    1. Let u=2xu = - 2 x.

      Then let du=2dxdu = - 2 dx and substitute du2- \frac{du}{2}:

      (eu2)du\int \left(- \frac{e^{u}}{2}\right)\, du

      1. The integral of a constant times a function is the constant times the integral of the function:

        False\text{False}

        1. The integral of the exponential function is itself.

          eudu=eu\int e^{u}\, du = e^{u}

        So, the result is: eu2- \frac{e^{u}}{2}

      Now substitute uu back in:

      e2x2- \frac{e^{- 2 x}}{2}

    So, the result is: e2x- e^{- 2 x}

  3. Now simplify:

    (2x+1)e2x\left(2 x + 1\right) e^{- 2 x}

  4. Add the constant of integration:

    (2x+1)e2x+constant\left(2 x + 1\right) e^{- 2 x}+ \mathrm{constant}


The answer is:

(2x+1)e2x+constant\left(2 x + 1\right) e^{- 2 x}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                                     
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 |       -2*x               -2*x    -2*x
 | -4*x*e     dx = C + 2*x*e     + e    
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/                                       
4xe2xdx=C+2xe2x+e2x\int - 4 x e^{- 2 x}\, dx = C + 2 x e^{- 2 x} + e^{- 2 x}
The graph
0.001.000.100.200.300.400.500.600.700.800.902-2
The answer [src]
        -2
-1 + 3*e  
1+3e2-1 + \frac{3}{e^{2}}
=
=
        -2
-1 + 3*e  
1+3e2-1 + \frac{3}{e^{2}}
-1 + 3*exp(-2)
Numerical answer [src]
-0.593994150290162
-0.593994150290162
The graph
Integral of -4*x*exp(-2*x) dx

    Use the examples entering the upper and lower limits of integration.