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1/(2+cosx)^2
Solving integrals/ 1/(2+cosx)^2

Integral of 1/(2+cosx)^2 dx

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  1                 
  /                 
 |                  
 |        1         
 |  ------------- dx
 |              2   
 |  (2 + cos(x))    
 |                  
/                   
0
$$\int\limits_{0}^{1} \frac{1}{\left(\cos{\left(x \right)} + 2\right)^{2}}\, dx$$ 
Integral(1/((2 + cos(x))^2), (x, 0, 1))
The answer (Indefinite) [src] 
                                                    /        /x   pi\       /  ___    /x\\\                   /        /x   pi\       /  ___    /x\\\
                                                    |        |- - --|       |\/ 3 *tan|-|||                   |        |- - --|       |\/ 3 *tan|-|||
  /                               /x\           ___ |        |2   2 |       |         \2/||       ___    2/x\ |        |2   2 |       |         \2/||
 |                           6*tan|-|      12*\/ 3 *|pi*floor|------| + atan|------------||   4*\/ 3 *tan |-|*|pi*floor|------| + atan|------------||
 |       1                        \2/               \        \  pi  /       \     3      //               \2/ \        \  pi  /       \     3      //
 | ------------- dx = C - -------------- + ------------------------------------------------ + -------------------------------------------------------
 |             2                    2/x\                              2/x\                                                   2/x\                    
 | (2 + cos(x))           27 + 9*tan |-|                    27 + 9*tan |-|                                         27 + 9*tan |-|                    
 |                                   \2/                               \2/                                                    \2/                    
/
$$\int \frac{1}{\left(\cos{\left(x \right)} + 2\right)^{2}}\, dx = C + \frac{4 \sqrt{3} \left(\operatorname{atan}{\left(\frac{\sqrt{3} \tan{\left(\frac{x}{2} \right)}}{3} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right) \tan^{2}{\left(\frac{x}{2} \right)}}{9 \tan^{2}{\left(\frac{x}{2} \right)} + 27} + \frac{12 \sqrt{3} \left(\operatorname{atan}{\left(\frac{\sqrt{3} \tan{\left(\frac{x}{2} \right)}}{3} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{9 \tan^{2}{\left(\frac{x}{2} \right)} + 27} - \frac{6 \tan{\left(\frac{x}{2} \right)}}{9 \tan^{2}{\left(\frac{x}{2} \right)} + 27}$$
Simplify
The graph
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The answer [src] 
                                           /          /  ___         \\                     /          /  ___         \\
                                       ___ |          |\/ 3 *tan(1/2)||       ___    2      |          |\/ 3 *tan(1/2)||
                            ___   12*\/ 3 *|-pi + atan|--------------||   4*\/ 3 *tan (1/2)*|-pi + atan|--------------||
     6*tan(1/2)      4*pi*\/ 3             \          \      3       //                     \          \      3       //
- ---------------- + ---------- + ------------------------------------- + ----------------------------------------------
            2            9                             2                                           2                    
  27 + 9*tan (1/2)                           27 + 9*tan (1/2)                            27 + 9*tan (1/2)
$$\frac{12 \sqrt{3} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{3} \tan{\left(\frac{1}{2} \right)}}{3} \right)}\right)}{9 \tan^{2}{\left(\frac{1}{2} \right)} + 27} + \frac{4 \sqrt{3} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{3} \tan{\left(\frac{1}{2} \right)}}{3} \right)}\right) \tan^{2}{\left(\frac{1}{2} \right)}}{9 \tan^{2}{\left(\frac{1}{2} \right)} + 27} - \frac{6 \tan{\left(\frac{1}{2} \right)}}{9 \tan^{2}{\left(\frac{1}{2} \right)} + 27} + \frac{4 \sqrt{3} \pi}{9}$$ 
=
= 
                                           /          /  ___         \\                     /          /  ___         \\
                                       ___ |          |\/ 3 *tan(1/2)||       ___    2      |          |\/ 3 *tan(1/2)||
                            ___   12*\/ 3 *|-pi + atan|--------------||   4*\/ 3 *tan (1/2)*|-pi + atan|--------------||
     6*tan(1/2)      4*pi*\/ 3             \          \      3       //                     \          \      3       //
- ---------------- + ---------- + ------------------------------------- + ----------------------------------------------
            2            9                             2                                           2                    
  27 + 9*tan (1/2)                           27 + 9*tan (1/2)                            27 + 9*tan (1/2)
$$\frac{12 \sqrt{3} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{3} \tan{\left(\frac{1}{2} \right)}}{3} \right)}\right)}{9 \tan^{2}{\left(\frac{1}{2} \right)} + 27} + \frac{4 \sqrt{3} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{3} \tan{\left(\frac{1}{2} \right)}}{3} \right)}\right) \tan^{2}{\left(\frac{1}{2} \right)}}{9 \tan^{2}{\left(\frac{1}{2} \right)} + 27} - \frac{6 \tan{\left(\frac{1}{2} \right)}}{9 \tan^{2}{\left(\frac{1}{2} \right)} + 27} + \frac{4 \sqrt{3} \pi}{9}$$ 
-6*tan(1/2)/(27 + 9*tan(1/2)^2) + 4*pi*sqrt(3)/9 + 12*sqrt(3)*(-pi + atan(sqrt(3)*tan(1/2)/3))/(27 + 9*tan(1/2)^2) + 4*sqrt(3)*tan(1/2)^2*(-pi + atan(sqrt(3)*tan(1/2)/3))/(27 + 9*tan(1/2)^2)
Numerical answer [src] 
0.124782407257615
0.124782407257615
The graph
Integral of 1/(2+cosx)^2 dx

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

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