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(x+1)e^x

Integral of (x+1)e^x dx

Limits of integration:

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Piecewise:

The solution

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01ex(x+1)dx\int\limits_{0}^{1} e^{x} \left(x + 1\right)\, dx
Integral((x + 1)*E^x, (x, 0, 1))
Detail solution
  1. Rewrite the integrand:

    ex(x+1)=xex+exe^{x} \left(x + 1\right) = x e^{x} + e^{x}

  2. 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(x)=xu{\left(x \right)} = x and let dv(x)=ex\operatorname{dv}{\left(x \right)} = e^{x}.

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

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

      1. The integral of the exponential function is itself.

        exdx=ex\int e^{x}\, dx = e^{x}

      Now evaluate the sub-integral.

    2. The integral of the exponential function is itself.

      exdx=ex\int e^{x}\, dx = e^{x}

    1. The integral of the exponential function is itself.

      exdx=ex\int e^{x}\, dx = e^{x}

    The result is: xexx e^{x}

  3. Add the constant of integration:

    xex+constantx e^{x}+ \mathrm{constant}


The answer is:

xex+constantx e^{x}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                        
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 | (x + 1)*E  dx = C + x*e 
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ex(x+1)dx=C+xex\int e^{x} \left(x + 1\right)\, dx = C + x e^{x}
The graph
0.001.000.100.200.300.400.500.600.700.800.90010
The answer [src]
E
ee
=
=
E
ee
E
Numerical answer [src]
2.71828182845905
2.71828182845905
The graph
Integral of (x+1)e^x dx

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