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(x+2)e^(-x)

Integral of (x+2)e^(-x) dx

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The solution

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  1               
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01ex(x+2)dx\int\limits_{0}^{1} e^{- x} \left(x + 2\right)\, dx
Integral((x + 2)*E^(-x), (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=xu = - x.

      Then let du=dxdu = - dx and substitute dudu:

      (ueu2eu)du\int \left(u e^{u} - 2 e^{u}\right)\, du

      1. Integrate term-by-term:

        1. Use integration by parts:

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

          Let u(u)=uu{\left(u \right)} = u and let dv(u)=eu\operatorname{dv}{\left(u \right)} = e^{u}.

          Then du(u)=1\operatorname{du}{\left(u \right)} = 1.

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

          1. The integral of the exponential function is itself.

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

          Now evaluate the sub-integral.

        2. The integral of the exponential function is itself.

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

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

          (2eu)du=2eudu\int \left(- 2 e^{u}\right)\, du = - 2 \int e^{u}\, du

          1. The integral of the exponential function is itself.

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

          So, the result is: 2eu- 2 e^{u}

        The result is: ueu3euu e^{u} - 3 e^{u}

      Now substitute uu back in:

      xex3ex- x e^{- x} - 3 e^{- x}

    Method #2

    1. Rewrite the integrand:

      ex(x+2)=xex+2exe^{- x} \left(x + 2\right) = x e^{- x} + 2 e^{- x}

    2. Integrate term-by-term:

      1. Let u=xu = - x.

        Then let du=dxdu = - dx and substitute dudu:

        ueudu\int u e^{u}\, du

        1. Use integration by parts:

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

          Let u(u)=uu{\left(u \right)} = u and let dv(u)=eu\operatorname{dv}{\left(u \right)} = e^{u}.

          Then du(u)=1\operatorname{du}{\left(u \right)} = 1.

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

          1. The integral of the exponential function is itself.

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

          Now evaluate the sub-integral.

        2. The integral of the exponential function is itself.

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

        Now substitute uu back in:

        xexex- x e^{- x} - e^{- x}

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

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

        1. Let u=xu = - x.

          Then let du=dxdu = - dx and substitute du- du:

          (eu)du\int \left(- e^{u}\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: eu- e^{u}

          Now substitute uu back in:

          ex- e^{- x}

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

      The result is: xex3ex- x e^{- x} - 3 e^{- x}

  2. Now simplify:

    (x+3)ex- \left(x + 3\right) e^{- x}

  3. Add the constant of integration:

    (x+3)ex+constant- \left(x + 3\right) e^{- x}+ \mathrm{constant}


The answer is:

(x+3)ex+constant- \left(x + 3\right) e^{- x}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                                  
 |                                   
 |          -x             -x      -x
 | (x + 2)*E   dx = C - 3*e   - x*e  
 |                                   
/                                    
ex(x+2)dx=Cxex3ex\int e^{- x} \left(x + 2\right)\, dx = C - x e^{- x} - 3 e^{- x}
The graph
0.001.000.100.200.300.400.500.600.700.800.905-5
The answer [src]
       -1
3 - 4*e  
34e3 - \frac{4}{e}
=
=
       -1
3 - 4*e  
34e3 - \frac{4}{e}
3 - 4*exp(-1)
Numerical answer [src]
1.52848223531423
1.52848223531423
The graph
Integral of (x+2)e^(-x) dx

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