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Integral of (sin(x))/(1-cos(x)2) dx

The solution

You have entered [src]

  1                
  /                
 |                 
 |     sin(x)      
 |  ------------ dx
 |  1 - cos(x)*2   
 |                 
/                  
0

$$\int\limits_{0}^{1} \frac{\sin{\left(x \right)}}{- 2 \cos{\left(x \right)} + 1}\, dx$$

Integral(sin(x)/(1 - cos(x)*2), (x, 0, 1))

Detail solution

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

$\frac{\log{\left(1 - 2 \cos{\left(x \right)} \right)}}{2}$
Add the constant of integration:

$\frac{\log{\left(1 - 2 \cos{\left(x \right)} \right)}}{2}+ \mathrm{constant}$

The answer is:

$\frac{\log{\left(1 - 2 \cos{\left(x \right)} \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                       
 |                                        
 |    sin(x)             log(1 - cos(x)*2)
 | ------------ dx = C + -----------------
 | 1 - cos(x)*2                  2        
 |                                        
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

log(2)   log(-1/2 + cos(1))
------ + ------------------
  2              2

$$\frac{\log{\left(- \frac{1}{2} + \cos{\left(1 \right)} \right)}}{2} + \frac{\log{\left(2 \right)}}{2}$$

log(2)   log(-1/2 + cos(1))
------ + ------------------
  2              2

$$\frac{\log{\left(- \frac{1}{2} + \cos{\left(1 \right)} \right)}}{2} + \frac{\log{\left(2 \right)}}{2}$$

log(2)/2 + log(-1/2 + cos(1))/2

Numerical answer [src]

-1.25909970676736

-1.25909970676736

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

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Identical expressions

Similar expressions

Integral of (sin(x))/(1-cos(x)2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2