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Similar expressions
(exp^x)/(2-exp^(2x))

Solving integrals/ (exp^x)/(2+exp^(2x))

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

The solution

You have entered [src]

  1            
  /            
 |             
 |      x      
 |     e       
 |  -------- dx
 |       2*x   
 |  2 + e      
 |             
/              
0

\int\limits_{0}^{1} \frac{e^{x}}{e^{2 x} + 2}\, dx

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

Detail solution

Let $u = e^{x}$ .

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

$\int \frac{1}{u^{2} + 2}\, du$
1. The integral of $\frac{1}{u^{2} + 1}$ is $\frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} u}{2} \right)}}{2}$ .
Now substitute $u$ back in:

$\frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} e^{x}}{2} \right)}}{2}$
Add the constant of integration:

$\frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} e^{x}}{2} \right)}}{2}+ \mathrm{constant}$

The answer is:

\frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} e^{x}}{2} \right)}}{2}+ \mathrm{constant}

The answer (Indefinite) [src]

  /                            /  ___  x\
 |                     ___     |\/ 2 *e |
 |     x             \/ 2 *atan|--------|
 |    e                        \   2    /
 | -------- dx = C + --------------------
 |      2*x                   2          
 | 2 + e                                 
 |                                       
/

{{\arctan \left({{e^{x}}\over{\sqrt{2}}}\right)}\over{\sqrt{2}}}

Simplify

The graph

Plot the graph f(x)

The answer [src]

         /   2                         \          /   2                         \
- RootSum\8*z  + 1, i -> i*log(1 + 4*i)/ + RootSum\8*z  + 1, i -> i*log(e + 4*i)/

{{\arctan \left({{e}\over{\sqrt{2}}}\right)}\over{\sqrt{2}}}-{{ \arctan \left({{1}\over{\sqrt{2}}}\right)}\over{\sqrt{2}}}

         /   2                         \          /   2                         \
- RootSum\8*z  + 1, i -> i*log(1 + 4*i)/ + RootSum\8*z  + 1, i -> i*log(e + 4*i)/

- \operatorname{RootSum} {\left(8 z^{2} + 1, \left( i \mapsto i \log{\left(4 i + 1 \right)} \right)\right)} + \operatorname{RootSum} {\left(8 z^{2} + 1, \left( i \mapsto i \log{\left(4 i + e \right)} \right)\right)}

Simplify

Numerical answer [src]

0.336294761863011

0.336294761863011

The graph

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

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Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution