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1/(2+cos(x))

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

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The solution

You have entered [src]
  1              
  /              
 |               
 |      1        
 |  ---------- dx
 |  2 + cos(x)   
 |               
/                
0                
$$\int\limits_{0}^{1} \frac{1}{\cos{\left(x \right)} + 2}\, dx$$
Integral(1/(2 + cos(x)), (x, 0, 1))
The answer (Indefinite) [src]
                               /        /x   pi\       /  ___    /x\\\
                               |        |- - --|       |\/ 3 *tan|-|||
  /                        ___ |        |2   2 |       |         \2/||
 |                     2*\/ 3 *|pi*floor|------| + atan|------------||
 |     1                       \        \  pi  /       \     3      //
 | ---------- dx = C + -----------------------------------------------
 | 2 + cos(x)                                 3                       
 |                                                                    
/                                                                     
$$\int \frac{1}{\cos{\left(x \right)} + 2}\, dx = C + \frac{2 \sqrt{3} \left(\operatorname{atan}{\left(\frac{\sqrt{3} \tan{\left(\frac{x}{2} \right)}}{3} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{3}$$
The graph
The answer [src]
                     /          /  ___         \\
                 ___ |          |\/ 3 *tan(1/2)||
       ___   2*\/ 3 *|-pi + atan|--------------||
2*pi*\/ 3            \          \      3       //
---------- + ------------------------------------
    3                         3                  
$$\frac{2 \sqrt{3} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{3} \tan{\left(\frac{1}{2} \right)}}{3} \right)}\right)}{3} + \frac{2 \sqrt{3} \pi}{3}$$
=
=
                     /          /  ___         \\
                 ___ |          |\/ 3 *tan(1/2)||
       ___   2*\/ 3 *|-pi + atan|--------------||
2*pi*\/ 3            \          \      3       //
---------- + ------------------------------------
    3                         3                  
$$\frac{2 \sqrt{3} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{3} \tan{\left(\frac{1}{2} \right)}}{3} \right)}\right)}{3} + \frac{2 \sqrt{3} \pi}{3}$$
2*pi*sqrt(3)/3 + 2*sqrt(3)*(-pi + atan(sqrt(3)*tan(1/2)/3))/3
Numerical answer [src]
0.352797793265048
0.352797793265048
The graph
Integral of 1/(2+cos(x)) dx

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