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

The solution

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

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

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

Detail solution

Rewrite the integrand:

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

$\int \left(- \frac{1}{\cos{\left(x \right)} - 2}\right)\, dx = - \int \frac{1}{\cos{\left(x \right)} - 2}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $- \frac{2 \sqrt{3} \left(\operatorname{atan}{\left(\sqrt{3} \tan{\left(\frac{x}{2} \right)} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{3}$
So, the result is: $\frac{2 \sqrt{3} \left(\operatorname{atan}{\left(\sqrt{3} \tan{\left(\frac{x}{2} \right)} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{3}$
Now simplify:

$\frac{2 \sqrt{3} \left(\operatorname{atan}{\left(\sqrt{3} \tan{\left(\frac{x}{2} \right)} \right)} + \pi \left\lfloor{\frac{x}{2 \pi} - \frac{1}{2}}\right\rfloor\right)}{3}$
Add the constant of integration:

$\frac{2 \sqrt{3} \left(\operatorname{atan}{\left(\sqrt{3} \tan{\left(\frac{x}{2} \right)} \right)} + \pi \left\lfloor{\frac{x}{2 \pi} - \frac{1}{2}}\right\rfloor\right)}{3}+ \mathrm{constant}$

The answer is:

$\frac{2 \sqrt{3} \left(\operatorname{atan}{\left(\sqrt{3} \tan{\left(\frac{x}{2} \right)} \right)} + \pi \left\lfloor{\frac{x}{2 \pi} - \frac{1}{2}}\right\rfloor\right)}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

                               /        /x   pi\                     \
                               |        |- - --|                     |
  /                        ___ |        |2   2 |       /  ___    /x\\|
 |                     2*\/ 3 *|pi*floor|------| + atan|\/ 3 *tan|-|||
 |     1                       \        \  pi  /       \         \2///
 | ---------- dx = C + -----------------------------------------------
 | 2 - cos(x)                                 3                       
 |                                                                    
/

$$\int \frac{1}{2 - \cos{\left(x \right)}}\, dx = C + \frac{2 \sqrt{3} \left(\operatorname{atan}{\left(\sqrt{3} \tan{\left(\frac{x}{2} \right)} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

       ___       ___ /          /  ___         \\
2*pi*\/ 3    2*\/ 3 *\-pi + atan\\/ 3 *tan(1/2)//
---------- + ------------------------------------
    3                         3

$$\frac{2 \sqrt{3} \left(- \pi + \operatorname{atan}{\left(\sqrt{3} \tan{\left(\frac{1}{2} \right)} \right)}\right)}{3} + \frac{2 \sqrt{3} \pi}{3}$$

       ___       ___ /          /  ___         \\
2*pi*\/ 3    2*\/ 3 *\-pi + atan\\/ 3 *tan(1/2)//
---------- + ------------------------------------
    3                         3

$$\frac{2 \sqrt{3} \left(- \pi + \operatorname{atan}{\left(\sqrt{3} \tan{\left(\frac{1}{2} \right)} \right)}\right)}{3} + \frac{2 \sqrt{3} \pi}{3}$$

2*pi*sqrt(3)/3 + 2*sqrt(3)*(-pi + atan(sqrt(3)*tan(1/2)))/3

Numerical answer [src]

0.875002134035093

0.875002134035093

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

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Identical expressions

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

Limits of integration:

The graph:

Piecewise:

The solution