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ln(e^x+ one)
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Similar expressions
ln(e^x-1)

Integral of ln(e^x+1) dx

The solution

You have entered [src]

 2*pi              
   /               
  |                
  |     / x    \   
  |  log\E  + 1/ dx
  |                
 /                 
 0

$$\int\limits_{0}^{2 \pi} \log{\left(e^{x} + 1 \right)}\, dx$$

Integral(log(E^x + 1), (x, 0, 2*pi))

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = \log{\left(e^{x} + 1 \right)}$ and let $\operatorname{dv}{\left(x \right)} = 1$ .

Then $\operatorname{du}{\left(x \right)} = \frac{e^{x}}{e^{x} + 1}$ .

To find $v{\left(x \right)}$ :
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dx = x$
Now evaluate the sub-integral.
Don't know the steps in finding this integral.

But the integral is

$\int \frac{x e^{x}}{e^{x} + 1}\, dx$
Now simplify:

$x \log{\left(e^{x} + 1 \right)} - \int \frac{x e^{x}}{e^{x} + 1}\, dx$
Add the constant of integration:

$x \log{\left(e^{x} + 1 \right)} - \int \frac{x e^{x}}{e^{x} + 1}\, dx+ \mathrm{constant}$

The answer is:

$x \log{\left(e^{x} + 1 \right)} - \int \frac{x e^{x}}{e^{x} + 1}\, dx+ \mathrm{constant}$

The answer (Indefinite) [src]

                          /                         
  /                      |                          
 |                       |     x                    
 |    / x    \           |  x*e             / x    \
 | log\E  + 1/ dx = C -  | ------ dx + x*log\E  + 1/
 |                       |      x                   
/                        | 1 + e                    
                         |                          
                        /

$$\int \log{\left(e^{x} + 1 \right)}\, dx = C + x \log{\left(e^{x} + 1 \right)} - \int \frac{x e^{x}}{e^{x} + 1}\, dx$$

Simplify

The answer [src]

   2*pi                               
     /                                
    |                                 
    |      x                          
    |   x*e                /     2*pi\
-   |  ------ dx + 2*pi*log\1 + e    /
    |       x                         
    |  1 + e                          
    |                                 
   /                                  
   0

$$- \int\limits_{0}^{2 \pi} \frac{x e^{x}}{e^{x} + 1}\, dx + 2 \pi \log{\left(1 + e^{2 \pi} \right)}$$

   2*pi                               
     /                                
    |                                 
    |      x                          
    |   x*e                /     2*pi\
-   |  ------ dx + 2*pi*log\1 + e    /
    |       x                         
    |  1 + e                          
    |                                 
   /                                  
   0

$$- \int\limits_{0}^{2 \pi} \frac{x e^{x}}{e^{x} + 1}\, dx + 2 \pi \log{\left(1 + e^{2 \pi} \right)}$$

-Integral(x*exp(x)/(1 + exp(x)), (x, 0, 2*pi)) + 2*pi*log(1 + exp(2*pi))

Numerical answer [src]

20.5598092639839

20.5598092639839

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

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Integral of ln(e^x+1) dx

Limits of integration:

The graph:

Piecewise:

The solution