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e^x/(e^x+2)

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

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

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  1          
  /          
 |           
 |     x     
 |    E      
 |  ------ dx
 |   x       
 |  E  + 2   
 |           
/            
0            
01exex+2dx\int\limits_{0}^{1} \frac{e^{x}}{e^{x} + 2}\, dx
Integral(E^x/(E^x + 2), (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=exu = e^{x}.

      Then let du=exdxdu = e^{x} dx and substitute dudu:

      1u+2du\int \frac{1}{u + 2}\, du

      1. Let u=u+2u = u + 2.

        Then let du=dudu = du and substitute dudu:

        1udu\int \frac{1}{u}\, du

        1. The integral of 1u\frac{1}{u} is log(u)\log{\left(u \right)}.

        Now substitute uu back in:

        log(u+2)\log{\left(u + 2 \right)}

      Now substitute uu back in:

      log(ex+2)\log{\left(e^{x} + 2 \right)}

    Method #2

    1. Let u=ex+2u = e^{x} + 2.

      Then let du=exdxdu = e^{x} dx and substitute dudu:

      1udu\int \frac{1}{u}\, du

      1. The integral of 1u\frac{1}{u} is log(u)\log{\left(u \right)}.

      Now substitute uu back in:

      log(ex+2)\log{\left(e^{x} + 2 \right)}

  2. Now simplify:

    log(ex+2)\log{\left(e^{x} + 2 \right)}

  3. Add the constant of integration:

    log(ex+2)+constant\log{\left(e^{x} + 2 \right)}+ \mathrm{constant}


The answer is:

log(ex+2)+constant\log{\left(e^{x} + 2 \right)}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                           
 |                            
 |    x                       
 |   E                /     x\
 | ------ dx = C + log\2 + E /
 |  x                         
 | E  + 2                     
 |                            
/                             
exex+2dx=C+log(ex+2)\int \frac{e^{x}}{e^{x} + 2}\, dx = C + \log{\left(e^{x} + 2 \right)}
The graph
0.001.000.100.200.300.400.500.600.700.800.9002
The answer [src]
-log(3) + log(2 + E)
log(3)+log(2+e)- \log{\left(3 \right)} + \log{\left(2 + e \right)}
=
=
-log(3) + log(2 + E)
log(3)+log(2+e)- \log{\left(3 \right)} + \log{\left(2 + e \right)}
-log(3) + log(2 + E)
Numerical answer [src]
0.452832425263941
0.452832425263941
The graph
Integral of e^x/(e^x+2) dx

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