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Solving integrals/ e^x/(e^x-2)

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  3          
  /          
 |           
 |     x     
 |    E      
 |  ------ dx
 |   x       
 |  E  - 2   
 |           
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0
∫03exex−2 dx\int\limits_{0}^{3} \frac{e^{x}}{e^{x} - 2}\, dx 
Integral(E^x/(E^x - 2), (x, 0, 3))
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−2 du\int \frac{1}{u - 2}\, du

      1. Let u=u−2u = u - 2.

        Then let du=dudu = du and substitute dudu:

        ∫1u du\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:

      ∫1u du\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−2 dx=C+log⁡(ex−2)\int \frac{e^{x}}{e^{x} - 2}\, dx = C + \log{\left(e^{x} - 2 \right)}
Simplify
The graph
0.003.000.250.500.751.001.251.501.752.002.252.502.75-1000010000
Plot the graph f(x)
The answer [src] 
nan
NaN\text{NaN} 
=
= 
nan
NaN\text{NaN} 
nan
Numerical answer [src] 
-11.8764540370141
-11.8764540370141

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

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