Integral of x/(1-cos2x) dx

The solution

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

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

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

Detail solution

Rewrite the integrand:

$\frac{x}{1 - \cos{\left(2 x \right)}} = - \frac{x}{\cos{\left(2 x \right)} - 1}$
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- \frac{x}{\cos{\left(2 x \right)} - 1}\right)\, dx = - \int \frac{x}{\cos{\left(2 x \right)} - 1}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\frac{x}{2 \tan{\left(x \right)}} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{4} - \frac{\log{\left(\tan{\left(x \right)} \right)}}{2}$
So, the result is: $- \frac{x}{2 \tan{\left(x \right)}} - \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{4} + \frac{\log{\left(\tan{\left(x \right)} \right)}}{2}$
Now simplify:

$- \frac{x}{2 \tan{\left(x \right)}} - \frac{\log{\left(\frac{1}{\cos^{2}{\left(x \right)}} \right)}}{4} + \frac{\log{\left(\tan{\left(x \right)} \right)}}{2}$
Add the constant of integration:

$- \frac{x}{2 \tan{\left(x \right)}} - \frac{\log{\left(\frac{1}{\cos^{2}{\left(x \right)}} \right)}}{4} + \frac{\log{\left(\tan{\left(x \right)} \right)}}{2}+ \mathrm{constant}$

The answer is:

$- \frac{x}{2 \tan{\left(x \right)}} - \frac{\log{\left(\frac{1}{\cos^{2}{\left(x \right)}} \right)}}{4} + \frac{\log{\left(\tan{\left(x \right)} \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                               
 |                                        /       2   \           
 |      x                log(tan(x))   log\1 + tan (x)/      x    
 | ------------ dx = C + ----------- - ---------------- - --------
 | 1 - cos(2*x)               2               4           2*tan(x)
 |                                                                
/

$$\int \frac{x}{1 - \cos{\left(2 x \right)}}\, dx = C - \frac{x}{2 \tan{\left(x \right)}} - \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{4} + \frac{\log{\left(\tan{\left(x \right)} \right)}}{2}$$

Simplify

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Plot the graph f(x)

The answer [src]

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The graph

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

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Integral of x/(1-cos2x) dx

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The solution