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

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

The solution

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

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

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

Detail solution

Rewrite the integrand:

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

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

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$$\infty$$

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$$\infty$$

Упростить

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

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

Limits of integration:

The graph:

Piecewise:

The solution