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Identical expressions
(exp(x)- one)/(one -cos(x))
( exponent of (x) minus 1) divide by (1 minus co sinus of e of (x))
( exponent of (x) minus one) divide by (one minus co sinus of e of (x))
expx-1/1-cosx
(exp(x)-1) divide by (1-cos(x))
(exp(x)-1)/(1-cos(x))dx
Similar expressions
(exp(x)-1)/(1+cos(x))
(exp(x)+1)/(1-cos(x))
(exp(x)-1)/(1-cosx)

Solving integrals/ (exp(x)-1)/(1-cos(x))

Integral of (exp(x)-1)/(1-cos(x)) dx

The solution

You have entered [src]

  1              
  /              
 |               
 |     x         
 |    e  - 1     
 |  ---------- dx
 |  1 - cos(x)   
 |               
/                
0

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

Integral((exp(x) - 1)/(1 - cos(x)), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{e^{x} - 1}{1 - \cos{\left(x \right)}} = - \frac{e^{x} - 1}{\cos{\left(x \right)} - 1}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{e^{x} - 1}{\cos{\left(x \right)} - 1}\right)\, dx = - \int \frac{e^{x} - 1}{\cos{\left(x \right)} - 1}\, dx$
  1. Rewrite the integrand:
    
    $\frac{e^{x} - 1}{\cos{\left(x \right)} - 1} = \frac{e^{x}}{\cos{\left(x \right)} - 1} - \frac{1}{\cos{\left(x \right)} - 1}$
  2. Integrate term-by-term:
    1. Don't know the steps in finding this integral.
      
      But the integral is
      
      $\int \frac{e^{x}}{\cos{\left(x \right)} - 1}\, dx$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{\cos{\left(x \right)} - 1}\right)\, dx = - \int \frac{1}{\cos{\left(x \right)} - 1}\, dx$
      
      Don't know the steps in finding this integral.
      
      But the integral is
      
      $\frac{1}{\tan{\left(\frac{x}{2} \right)}}$
      
      So, the result is: $- \frac{1}{\tan{\left(\frac{x}{2} \right)}}$
    The result is: $\int \frac{e^{x}}{\cos{\left(x \right)} - 1}\, dx - \frac{1}{\tan{\left(\frac{x}{2} \right)}}$
  So, the result is: $- \int \frac{e^{x}}{\cos{\left(x \right)} - 1}\, dx + \frac{1}{\tan{\left(\frac{x}{2} \right)}}$
Method #2
1. Rewrite the integrand:
  
  $\frac{e^{x} - 1}{1 - \cos{\left(x \right)}} = \frac{e^{x}}{1 - \cos{\left(x \right)}} - \frac{1}{1 - \cos{\left(x \right)}}$
2. Integrate term-by-term:
  1. Rewrite the integrand:
    
    $\frac{e^{x}}{1 - \cos{\left(x \right)}} = - \frac{e^{x}}{\cos{\left(x \right)} - 1}$
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{e^{x}}{\cos{\left(x \right)} - 1}\right)\, dx = - \int \frac{e^{x}}{\cos{\left(x \right)} - 1}\, dx$
    1. Don't know the steps in finding this integral.
      
      But the integral is
      
      $\int \frac{e^{x}}{\cos{\left(x \right)} - 1}\, dx$
    So, the result is: $- \int \frac{e^{x}}{\cos{\left(x \right)} - 1}\, dx$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{1}{1 - \cos{\left(x \right)}}\right)\, dx = - \int \frac{1}{1 - \cos{\left(x \right)}}\, dx$
    1. Rewrite the integrand:
      
      $\frac{1}{1 - \cos{\left(x \right)}} = - \frac{1}{\cos{\left(x \right)} - 1}$
    2. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{\cos{\left(x \right)} - 1}\right)\, dx = - \int \frac{1}{\cos{\left(x \right)} - 1}\, dx$
      
      Don't know the steps in finding this integral.
      
      But the integral is
      
      $\frac{1}{\tan{\left(\frac{x}{2} \right)}}$
      
      So, the result is: $- \frac{1}{\tan{\left(\frac{x}{2} \right)}}$
    So, the result is: $\frac{1}{\tan{\left(\frac{x}{2} \right)}}$
  The result is: $- \int \frac{e^{x}}{\cos{\left(x \right)} - 1}\, dx + \frac{1}{\tan{\left(\frac{x}{2} \right)}}$
Add the constant of integration:

$- \int \frac{e^{x}}{\cos{\left(x \right)} - 1}\, dx + \frac{1}{\tan{\left(\frac{x}{2} \right)}}+ \mathrm{constant}$

The answer is:

$- \int \frac{e^{x}}{\cos{\left(x \right)} - 1}\, dx + \frac{1}{\tan{\left(\frac{x}{2} \right)}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                               /              
 |                               |               
 |    x                          |       x       
 |   e  - 1              1       |      e        
 | ---------- dx = C + ------ -  | ----------- dx
 | 1 - cos(x)             /x\    | -1 + cos(x)   
 |                     tan|-|    |               
/                         \2/   /

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

Simplify

The answer [src]

                         1               
    1                    /               
    /                   |                
   |                    |        x       
   |      -1            |       e        
-  |  ----------- dx -  |  ----------- dx
   |  -1 + cos(x)       |  -1 + cos(x)   
   |                    |                
  /                    /                 
  0                    0

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

                         1               
    1                    /               
    /                   |                
   |                    |        x       
   |      -1            |       e        
-  |  ----------- dx -  |  ----------- dx
   |  -1 + cos(x)       |  -1 + cos(x)   
   |                    |                
  /                    /                 
  0                    0

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

-Integral(-1/(-1 + cos(x)), (x, 0, 1)) - Integral(exp(x)/(-1 + cos(x)), (x, 0, 1))

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

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Identical expressions

Similar expressions

Integral of (exp(x)-1)/(1-cos(x)) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2