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Similar expressions
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Solving integrals/ 1/(exp^x+exp^-x)

Integral of 1/(exp^x+exp^-x) dx

The solution

You have entered [src]

 log(3)             
    /               
   |                
   |        1       
   |   1*-------- dx
   |      x    -x   
   |     e  + e     
   |                
  /                 
  0

$$\int\limits_{0}^{\log{\left(3 \right)}} 1 \cdot \frac{1}{e^{x} + e^{- x}}\, dx$$

Integral(1/(E^x + E^(-x)), (x, 0, log(3)))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = e^{x}$ .
  
  Then let $du = e^{x} dx$ and substitute $du$ :
  
  $\int \frac{1}{u^{2} + 1}\, du$
  1. The integral of $\frac{1}{u^{2} + 1}$ is $\operatorname{atan}{\left(u \right)}$ .
  Now substitute $u$ back in:
  
  $\operatorname{atan}{\left(e^{x} \right)}$
Method #2
1. Rewrite the integrand:
  
  $1 \cdot \frac{1}{e^{x} + e^{- x}} = \frac{e^{x}}{e^{2 x} + 1}$
2. Let $u = e^{x}$ .
  
  Then let $du = e^{x} dx$ and substitute $du$ :
  
  $\int \frac{1}{u^{2} + 1}\, du$
  1. The integral of $\frac{1}{u^{2} + 1}$ is $\operatorname{atan}{\left(u \right)}$ .
  Now substitute $u$ back in:
  
  $\operatorname{atan}{\left(e^{x} \right)}$
Now simplify:

$\operatorname{atan}{\left(e^{x} \right)}$
Add the constant of integration:

$\operatorname{atan}{\left(e^{x} \right)}+ \mathrm{constant}$

The answer is:

$\operatorname{atan}{\left(e^{x} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

$$-\arctan e^ {- x }$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

         /   2                         \          /   2                         \
- RootSum\4*z  + 1, i -> i*log(1 + 2*i)/ + RootSum\4*z  + 1, i -> i*log(3 + 2*i)/

$${{\pi}\over{4}}-\arctan \left({{1}\over{3}}\right)$$

         /   2                         \          /   2                         \
- RootSum\4*z  + 1, i -> i*log(1 + 2*i)/ + RootSum\4*z  + 1, i -> i*log(3 + 2*i)/

$$- \operatorname{RootSum} {\left(4 z^{2} + 1, \left( i \mapsto i \log{\left(2 i + 1 \right)} \right)\right)} + \operatorname{RootSum} {\left(4 z^{2} + 1, \left( i \mapsto i \log{\left(2 i + 3 \right)} \right)\right)}$$

Упростить

Numerical answer [src]

0.463647609000806

0.463647609000806

The graph

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

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Identical expressions

Similar expressions

Integral of 1/(exp^x+exp^-x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2