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Resolución de integrales/ cos²x/ 1/(1+cos²x)

Integral de 1/(1+cos²x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1               
  /               
 |                
 |       1        
 |  ----------- dx
 |         2      
 |  1 + cos (x)   
 |                
/                 
0
∫011cos⁡2(x)+1 dx\int\limits_{0}^{1} \frac{1}{\cos^{2}{\left(x \right)} + 1}\, dx 
Integral(1/(1 + cos(x)^2), (x, 0, 1))
Respuesta (Indefinida) [src] 
                              /        /x   pi\                         \         /        /x   pi\                          \
                              |        |- - --|                         |         |        |- - --|                          |
  /                       ___ |        |2   2 |       /      ___    /x\\|     ___ |        |2   2 |       /       ___    /x\\|
 |                      \/ 2 *|pi*floor|------| + atan|1 + \/ 2 *tan|-|||   \/ 2 *|pi*floor|------| + atan|-1 + \/ 2 *tan|-|||
 |      1                     \        \  pi  /       \             \2///         \        \  pi  /       \              \2///
 | ----------- dx = C + ------------------------------------------------- + --------------------------------------------------
 |        2                                     2                                                   2                         
 | 1 + cos (x)                                                                                                                
 |                                                                                                                            
/
∫1cos⁡2(x)+1 dx=C+2(atan⁡(2tan⁡(x2)−1)+π⌊x2−π2π⌋)2+2(atan⁡(2tan⁡(x2)+1)+π⌊x2−π2π⌋)2\int \frac{1}{\cos^{2}{\left(x \right)} + 1}\, dx = C + \frac{\sqrt{2} \left(\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{2} + \frac{\sqrt{2} \left(\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{2}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.905-5
Trazar el gráfico f(x)
Respuesta [src] 
             ___ /          /      ___         \\     ___ /          /      ___         \\
     ___   \/ 2 *\-pi - atan\1 - \/ 2 *tan(1/2)//   \/ 2 *\-pi + atan\1 + \/ 2 *tan(1/2)//
pi*\/ 2  + -------------------------------------- + --------------------------------------
                             2                                        2
2(−π−atan⁡(−2tan⁡(12)+1))2+2(−π+atan⁡(2tan⁡(12)+1))2+2π\frac{\sqrt{2} \left(- \pi - \operatorname{atan}{\left(- \sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right)}{2} + \frac{\sqrt{2} \left(- \pi + \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right)}{2} + \sqrt{2} \pi 
=
= 
             ___ /          /      ___         \\     ___ /          /      ___         \\
     ___   \/ 2 *\-pi - atan\1 - \/ 2 *tan(1/2)//   \/ 2 *\-pi + atan\1 + \/ 2 *tan(1/2)//
pi*\/ 2  + -------------------------------------- + --------------------------------------
                             2                                        2
2(−π−atan⁡(−2tan⁡(12)+1))2+2(−π+atan⁡(2tan⁡(12)+1))2+2π\frac{\sqrt{2} \left(- \pi - \operatorname{atan}{\left(- \sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right)}{2} + \frac{\sqrt{2} \left(- \pi + \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right)}{2} + \sqrt{2} \pi 
pi*sqrt(2) + sqrt(2)*(-pi - atan(1 - sqrt(2)*tan(1/2)))/2 + sqrt(2)*(-pi + atan(1 + sqrt(2)*tan(1/2)))/2
Respuesta numérica [src] 
0.589407539463719
0.589407539463719

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

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