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Resolución de integrales/ cos(x)/ 1/(1+4(cos(x))^2)

Integral de 1/(1+4(cos(x))^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  0                 
  /                 
 |                  
 |        1         
 |  ------------- dx
 |           2      
 |  1 + 4*cos (x)   
 |                  
/                   
pi                  
--                  
4
∫π4014cos⁡2(x)+1 dx\int\limits_{\frac{\pi}{4}}^{0} \frac{1}{4 \cos^{2}{\left(x \right)} + 1}\, dx 
Integral(1/(1 + 4*cos(x)^2), (x, pi/4, 0))
Respuesta (Indefinida) [src] 
                                /        /x   pi\                          \         /        /x   pi\                         \
                                |        |- - --|                          |         |        |- - --|                         |
  /                         ___ |        |2   2 |       /       ___    /x\\|     ___ |        |2   2 |       /      ___    /x\\|
 |                        \/ 5 *|pi*floor|------| + atan|-2 + \/ 5 *tan|-|||   \/ 5 *|pi*floor|------| + atan|2 + \/ 5 *tan|-|||
 |       1                      \        \  pi  /       \              \2///         \        \  pi  /       \             \2///
 | ------------- dx = C + -------------------------------------------------- + -------------------------------------------------
 |          2                                     5                                                    5                        
 | 1 + 4*cos (x)                                                                                                                
 |                                                                                                                              
/
∫14cos⁡2(x)+1 dx=C+5(atan⁡(5tan⁡(x2)−2)+π⌊x2−π2π⌋)5+5(atan⁡(5tan⁡(x2)+2)+π⌊x2−π2π⌋)5\int \frac{1}{4 \cos^{2}{\left(x \right)} + 1}\, dx = C + \frac{\sqrt{5} \left(\operatorname{atan}{\left(\sqrt{5} \tan{\left(\frac{x}{2} \right)} - 2 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{5} + \frac{\sqrt{5} \left(\operatorname{atan}{\left(\sqrt{5} \tan{\left(\frac{x}{2} \right)} + 2 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{5}
Simplificar
Gráfica
0.000.050.100.150.200.250.300.350.400.450.500.550.600.650.700.755-5
Trazar el gráfico f(x)
Respuesta [src] 
    ___ /          /      ___ /       ___\\\     ___ /          /      ___ /       ___\\\     ___                     ___                
  \/ 5 *\-pi - atan\2 - \/ 5 *\-1 + \/ 2 ///   \/ 5 *\-pi + atan\2 + \/ 5 *\-1 + \/ 2 ///   \/ 5 *(-pi - atan(2))   \/ 5 *(-pi + atan(2))
- ------------------------------------------ - ------------------------------------------ + --------------------- + ---------------------
                      5                                            5                                  5                       5
5(−π−atan⁡(2))5+5(−π+atan⁡(2))5−5(−π+atan⁡(5(−1+2)+2))5−5(−π−atan⁡(−5(−1+2)+2))5\frac{\sqrt{5} \left(- \pi - \operatorname{atan}{\left(2 \right)}\right)}{5} + \frac{\sqrt{5} \left(- \pi + \operatorname{atan}{\left(2 \right)}\right)}{5} - \frac{\sqrt{5} \left(- \pi + \operatorname{atan}{\left(\sqrt{5} \left(-1 + \sqrt{2}\right) + 2 \right)}\right)}{5} - \frac{\sqrt{5} \left(- \pi - \operatorname{atan}{\left(- \sqrt{5} \left(-1 + \sqrt{2}\right) + 2 \right)}\right)}{5} 
=
= 
    ___ /          /      ___ /       ___\\\     ___ /          /      ___ /       ___\\\     ___                     ___                
  \/ 5 *\-pi - atan\2 - \/ 5 *\-1 + \/ 2 ///   \/ 5 *\-pi + atan\2 + \/ 5 *\-1 + \/ 2 ///   \/ 5 *(-pi - atan(2))   \/ 5 *(-pi + atan(2))
- ------------------------------------------ - ------------------------------------------ + --------------------- + ---------------------
                      5                                            5                                  5                       5
5(−π−atan⁡(2))5+5(−π+atan⁡(2))5−5(−π+atan⁡(5(−1+2)+2))5−5(−π−atan⁡(−5(−1+2)+2))5\frac{\sqrt{5} \left(- \pi - \operatorname{atan}{\left(2 \right)}\right)}{5} + \frac{\sqrt{5} \left(- \pi + \operatorname{atan}{\left(2 \right)}\right)}{5} - \frac{\sqrt{5} \left(- \pi + \operatorname{atan}{\left(\sqrt{5} \left(-1 + \sqrt{2}\right) + 2 \right)}\right)}{5} - \frac{\sqrt{5} \left(- \pi - \operatorname{atan}{\left(- \sqrt{5} \left(-1 + \sqrt{2}\right) + 2 \right)}\right)}{5} 
-sqrt(5)*(-pi - atan(2 - sqrt(5)*(-1 + sqrt(2))))/5 - sqrt(5)*(-pi + atan(2 + sqrt(5)*(-1 + sqrt(2))))/5 + sqrt(5)*(-pi - atan(2))/5 + sqrt(5)*(-pi + atan(2))/5
Respuesta numérica [src] 
-0.188068672113527
-0.188068672113527

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

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