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Limit of the function:
(1-cos(x))/(x-sin(x))
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(1-cos(x))/(x-sin(x))dx
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Solving integrals/ (1-cos(x))/(x-sin(x))

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

The solution

You have entered [src]

 2*pi             
   /              
  |               
  |  1 - cos(x)   
  |  ---------- dx
  |  x - sin(x)   
  |               
 /                
 pi

$$\int\limits_{\pi}^{2 \pi} \frac{1 - \cos{\left(x \right)}}{x - \sin{\left(x \right)}}\, dx$$

Integral((1 - cos(x))/(x - sin(x)), (x, pi, 2*pi))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = x - \sin{\left(x \right)}$ .
  
  Then let $du = \left(1 - \cos{\left(x \right)}\right) dx$ and substitute $du$ :
  
  $\int \frac{1}{u}\, du$
  1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
  Now substitute $u$ back in:
  
  $\log{\left(x - \sin{\left(x \right)} \right)}$
Method #2
1. Rewrite the integrand:
  
  $\frac{1 - \cos{\left(x \right)}}{x - \sin{\left(x \right)}} = - \frac{\cos{\left(x \right)} - 1}{x - \sin{\left(x \right)}}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{\cos{\left(x \right)} - 1}{x - \sin{\left(x \right)}}\right)\, dx = - \int \frac{\cos{\left(x \right)} - 1}{x - \sin{\left(x \right)}}\, dx$
  1. Let $u = x - \sin{\left(x \right)}$ .
    
    Then let $du = \left(1 - \cos{\left(x \right)}\right) dx$ and substitute $- du$ :
    
    $\int \left(- \frac{1}{u}\right)\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{u}\, du = - \int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $- \log{\left(u \right)}$
    Now substitute $u$ back in:
    
    $- \log{\left(x - \sin{\left(x \right)} \right)}$
  So, the result is: $\log{\left(x - \sin{\left(x \right)} \right)}$
Add the constant of integration:

$\log{\left(x - \sin{\left(x \right)} \right)}+ \mathrm{constant}$

The answer is:

$\log{\left(x - \sin{\left(x \right)} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                   
 |                                    
 | 1 - cos(x)                         
 | ---------- dx = C + log(x - sin(x))
 | x - sin(x)                         
 |                                    
/

$$\int \frac{1 - \cos{\left(x \right)}}{x - \sin{\left(x \right)}}\, dx = C + \log{\left(x - \sin{\left(x \right)} \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-log(pi) + log(2*pi)

$$- \log{\left(\pi \right)} + \log{\left(2 \pi \right)}$$

-log(pi) + log(2*pi)

$$- \log{\left(\pi \right)} + \log{\left(2 \pi \right)}$$

-log(pi) + log(2*pi)

Numerical answer [src]

0.693147180559945

0.693147180559945

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

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Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2