Mister Exam

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Integral of d{x}:
Integral of e^(-x)
Integral of -sin(x)
Integral of cos(x/2)
Integral of x-2
Identical expressions
(e^x-x)/((two *x))
(e to the power of x minus x) divide by ((2 multiply by x))
(e to the power of x minus x) divide by ((two multiply by x))
(ex-x)/((2*x))
ex-x/2*x
(e^x-x)/((2x))
(ex-x)/((2x))
ex-x/2x
e^x-x/2x
(e^x-x) divide by ((2*x))
(e^x-x)/((2*x))dx
Similar expressions
(e^x+x)/((2*x))

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

The solution

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  1          
  /          
 |           
 |   x       
 |  E  - x   
 |  ------ dx
 |   2*x     
 |           
/            
0

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

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

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = - x$ .
  
  Then let $du = - dx$ and substitute $\frac{du}{2}$ :
  
  $\int \frac{\left(u e^{u} + 1\right) e^{- u}}{2 u}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\left(u e^{u} + 1\right) e^{- u}}{u}\, du = \frac{\int \frac{\left(u e^{u} + 1\right) e^{- u}}{u}\, du}{2}$
    1. Let $u = \frac{1}{u}$ .
      
      Then let $du = - \frac{du}{u^{2}}$ and substitute $- du$ :
      
      $\int \left(- \frac{\left(u + e^{\frac{1}{u}}\right) e^{- \frac{1}{u}}}{u^{2}}\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\left(u + e^{\frac{1}{u}}\right) e^{- \frac{1}{u}}}{u^{2}}\, du = - \int \frac{\left(u + e^{\frac{1}{u}}\right) e^{- \frac{1}{u}}}{u^{2}}\, du$
      
      Rewrite the integrand:
      
      $\frac{\left(u + e^{\frac{1}{u}}\right) e^{- \frac{1}{u}}}{u^{2}} = \frac{e^{- \frac{1}{u}}}{u} + \frac{1}{u^{2}}$
      
      Integrate term-by-term:
      
      Let $u = \frac{1}{u}$ .
      
      Then let $du = - \frac{du}{u^{2}}$ and substitute $- du$ :
      
      $\int \left(- \frac{e^{- u}}{u}\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{e^{- u}}{u}\, du = - \int \frac{e^{- u}}{u}\, du$
      
      EiRule(a=-1, b=0, context=exp(-_u)/_u, symbol=_u)
      
      So, the result is: $- \operatorname{Ei}{\left(- u \right)}$
      
      Now substitute $u$ back in:
      
      $- \operatorname{Ei}{\left(- \frac{1}{u} \right)}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{2}}\, du = - \frac{1}{u}$
      
      The result is: $- \operatorname{Ei}{\left(- \frac{1}{u} \right)} - \frac{1}{u}$
      
      So, the result is: $\operatorname{Ei}{\left(- \frac{1}{u} \right)} + \frac{1}{u}$
      
      Now substitute $u$ back in:
      
      $u + \operatorname{Ei}{\left(- u \right)}$
    So, the result is: $\frac{u}{2} + \frac{\operatorname{Ei}{\left(- u \right)}}{2}$
  Now substitute $u$ back in:
  
  $- \frac{x}{2} + \frac{\operatorname{Ei}{\left(x \right)}}{2}$
Method #2
1. Rewrite the integrand:
  
  $\frac{e^{x} - x}{2 x} = - \frac{x - e^{x}}{2 x}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{x - e^{x}}{2 x}\right)\, dx = - \frac{\int \frac{x - e^{x}}{x}\, dx}{2}$
  1. Let $u = \frac{1}{x}$ .
    
    Then let $du = - \frac{dx}{x^{2}}$ and substitute $du$ :
    
    $\int \frac{u e^{\frac{1}{u}} - 1}{u^{2}}\, du$
    1. Rewrite the integrand:
      
      $\frac{u e^{\frac{1}{u}} - 1}{u^{2}} = \frac{e^{\frac{1}{u}}}{u} - \frac{1}{u^{2}}$
    2. Integrate term-by-term:
      
      Let $u = \frac{1}{u}$ .
      
      Then let $du = - \frac{du}{u^{2}}$ and substitute $- du$ :
      
      $\int \left(- \frac{e^{u}}{u}\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{e^{u}}{u}\, du = - \int \frac{e^{u}}{u}\, du$
      
      EiRule(a=1, b=0, context=exp(_u)/_u, symbol=_u)
      
      So, the result is: $- \operatorname{Ei}{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $- \operatorname{Ei}{\left(\frac{1}{u} \right)}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{u^{2}}\right)\, du = - \int \frac{1}{u^{2}}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{2}}\, du = - \frac{1}{u}$
      
      So, the result is: $\frac{1}{u}$
      
      The result is: $- \operatorname{Ei}{\left(\frac{1}{u} \right)} + \frac{1}{u}$
    Now substitute $u$ back in:
    
    $x - \operatorname{Ei}{\left(x \right)}$
  So, the result is: $- \frac{x}{2} + \frac{\operatorname{Ei}{\left(x \right)}}{2}$
Method #3
1. Rewrite the integrand:
  
  $\frac{e^{x} - x}{2 x} = - \frac{1}{2} + \frac{e^{x}}{2 x}$
2. Integrate term-by-term:
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \left(- \frac{1}{2}\right)\, dx = - \frac{x}{2}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{e^{x}}{2 x}\, dx = \frac{\int \frac{e^{x}}{x}\, dx}{2}$
    So, the result is: $\frac{\operatorname{Ei}{\left(x \right)}}{2}$
  The result is: $- \frac{x}{2} + \frac{\operatorname{Ei}{\left(x \right)}}{2}$
Add the constant of integration:

$- \frac{x}{2} + \frac{\operatorname{Ei}{\left(x \right)}}{2}+ \mathrm{constant}$

The answer is:

$- \frac{x}{2} + \frac{\operatorname{Ei}{\left(x \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                         
 |                          
 |  x                       
 | E  - x          Ei(x)   x
 | ------ dx = C + ----- - -
 |  2*x              2     2
 |                          
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

oo

$$\infty$$

oo

$$\infty$$

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Numerical answer [src]

22.2041741427236

22.2041741427236

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

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Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3