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Solving integrals/ 1/e^x+e^-x

Integral of 1/e^x+e^-x dx

The solution

You have entered [src]

  1              
  /              
 |               
 |  /1     -x\   
 |  |-- + E  | dx
 |  | x      |   
 |  \E       /   
 |               
/                
0

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

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

Detail solution

Integrate term-by-term:
1. Let $u = - x$ .
  
  Then let $du = - dx$ and substitute $- du$ :
  
  $\int \left(- e^{u}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\text{False}$
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    So, the result is: $- e^{u}$
  Now substitute $u$ back in:
  
  $- e^{- x}$
1. Let $u = e^{x}$ .
  
  Then let $du = e^{x} dx$ and substitute $du$ :
  
  $\int \frac{1}{u^{2}}\, du$
  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int \frac{1}{u^{2}}\, du = - \frac{1}{u}$
  Now substitute $u$ back in:
  
  $- e^{- x}$
The result is: $- 2 e^{- x}$
Add the constant of integration:

$- 2 e^{- x}+ \mathrm{constant}$

The answer is:

$- 2 e^{- x}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                         
 |                          
 | /1     -x\             -x
 | |-- + E  | dx = C - 2*e  
 | | x      |               
 | \E       /               
 |                          
/

$$\int \left(e^{- x} + \frac{1}{e^{x}}\right)\, dx = C - 2 e^{- x}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

       -1
2 - 2*e

$$2 - \frac{2}{e}$$

       -1
2 - 2*e

$$2 - \frac{2}{e}$$

2 - 2*exp(-1)

Numerical answer [src]

1.26424111765712

1.26424111765712

The graph

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

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Identical expressions

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Integral of 1/e^x+e^-x dx

Limits of integration:

The graph:

Piecewise:

The solution