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Integral of (sinx*dx)/(1-cosx) dx

The solution

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

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

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

Detail solution

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

$\log{\left(1 - \cos{\left(x \right)} \right)}+ \mathrm{constant}$

The answer is:

$\log{\left(1 - \cos{\left(x \right)} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$0$$

Numerical answer [src]

0.0

0.0

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

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How to use it?

Identical expressions

Similar expressions

Integral of (sinx*dx)/(1-cosx) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2