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Resolución de integrales/ 1/(x*x-2*x*cos(l)+1)

Integral de 1/(x*x-2*x*cos(l)+1) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                        
  /                        
 |                         
 |           1             
 |  -------------------- dx
 |  x*x - 2*x*cos(l) + 1   
 |                         
/                          
0
∫011(xx−2xcos⁡(l))+1 dx\int\limits_{0}^{1} \frac{1}{\left(x x - 2 x \cos{\left(l \right)}\right) + 1}\, dx 
Integral(1/(x*x - 2*x*cos(l) + 1), (x, 0, 1))
Respuesta (Indefinida) [src] 
                                     /   x - cos(l)   \
                                 atan|----------------|
  /                                  |   _____________|
 |                                   |  /        2    |
 |          1                        \\/  1 - cos (l) /
 | -------------------- dx = C + ----------------------
 | x*x - 2*x*cos(l) + 1                _____________   
 |                                    /        2       
/                                   \/  1 - cos (l)
∫1(xx−2xcos⁡(l))+1 dx=C+atan⁡(x−cos⁡(l)1−cos⁡2(l))1−cos⁡2(l)\int \frac{1}{\left(x x - 2 x \cos{\left(l \right)}\right) + 1}\, dx = C + \frac{\operatorname{atan}{\left(\frac{x - \cos{\left(l \right)}}{\sqrt{1 - \cos^{2}{\left(l \right)}}} \right)}}{\sqrt{1 - \cos^{2}{\left(l \right)}}}
Simplificar
Respuesta [src] 
    ____________________________    /      ____________________________                ____________________________        \       ____________________________    /        ____________________________                ____________________________        \       ____________________________    /    ____________________________                ____________________________        \       ____________________________    /        ____________________________                ____________________________        \
   /             1                  |     /             1                             /             1                  2   |      /             1                  |       /             1                             /             1                  2   |      /             1                  |   /             1                             /             1                  2   |      /             1                  |       /             1                             /             1                  2   |
  /  -------------------------- *log|-   /  --------------------------  - cos(l) +   /  -------------------------- *cos (l)|     /  -------------------------- *log|1 +   /  --------------------------  - cos(l) -   /  -------------------------- *cos (l)|     /  -------------------------- *log|  /  --------------------------  - cos(l) -   /  -------------------------- *cos (l)|     /  -------------------------- *log|1 -   /  --------------------------  - cos(l) +   /  -------------------------- *cos (l)|
\/   (1 + cos(l))*(-1 + cos(l))     \  \/   (1 + cos(l))*(-1 + cos(l))             \/   (1 + cos(l))*(-1 + cos(l))         /   \/   (1 + cos(l))*(-1 + cos(l))     \    \/   (1 + cos(l))*(-1 + cos(l))             \/   (1 + cos(l))*(-1 + cos(l))         /   \/   (1 + cos(l))*(-1 + cos(l))     \\/   (1 + cos(l))*(-1 + cos(l))             \/   (1 + cos(l))*(-1 + cos(l))         /   \/   (1 + cos(l))*(-1 + cos(l))     \    \/   (1 + cos(l))*(-1 + cos(l))             \/   (1 + cos(l))*(-1 + cos(l))         /
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                                                             2                                                                                                                               2                                                                                                                              2                                                                                                                              2
−1(cos⁡(l)−1)(cos⁡(l)+1)log⁡(−1(cos⁡(l)−1)(cos⁡(l)+1)cos⁡2(l)+1(cos⁡(l)−1)(cos⁡(l)+1)−cos⁡(l))2+1(cos⁡(l)−1)(cos⁡(l)+1)log⁡(1(cos⁡(l)−1)(cos⁡(l)+1)cos⁡2(l)−1(cos⁡(l)−1)(cos⁡(l)+1)−cos⁡(l))2+1(cos⁡(l)−1)(cos⁡(l)+1)log⁡(−1(cos⁡(l)−1)(cos⁡(l)+1)cos⁡2(l)+1(cos⁡(l)−1)(cos⁡(l)+1)−cos⁡(l)+1)2−1(cos⁡(l)−1)(cos⁡(l)+1)log⁡(1(cos⁡(l)−1)(cos⁡(l)+1)cos⁡2(l)−1(cos⁡(l)−1)(cos⁡(l)+1)−cos⁡(l)+1)2- \frac{\sqrt{\frac{1}{\left(\cos{\left(l \right)} - 1\right) \left(\cos{\left(l \right)} + 1\right)}} \log{\left(- \sqrt{\frac{1}{\left(\cos{\left(l \right)} - 1\right) \left(\cos{\left(l \right)} + 1\right)}} \cos^{2}{\left(l \right)} + \sqrt{\frac{1}{\left(\cos{\left(l \right)} - 1\right) \left(\cos{\left(l \right)} + 1\right)}} - \cos{\left(l \right)} \right)}}{2} + \frac{\sqrt{\frac{1}{\left(\cos{\left(l \right)} - 1\right) \left(\cos{\left(l \right)} + 1\right)}} \log{\left(\sqrt{\frac{1}{\left(\cos{\left(l \right)} - 1\right) \left(\cos{\left(l \right)} + 1\right)}} \cos^{2}{\left(l \right)} - \sqrt{\frac{1}{\left(\cos{\left(l \right)} - 1\right) \left(\cos{\left(l \right)} + 1\right)}} - \cos{\left(l \right)} \right)}}{2} + \frac{\sqrt{\frac{1}{\left(\cos{\left(l \right)} - 1\right) \left(\cos{\left(l \right)} + 1\right)}} \log{\left(- \sqrt{\frac{1}{\left(\cos{\left(l \right)} - 1\right) \left(\cos{\left(l \right)} + 1\right)}} \cos^{2}{\left(l \right)} + \sqrt{\frac{1}{\left(\cos{\left(l \right)} - 1\right) \left(\cos{\left(l \right)} + 1\right)}} - \cos{\left(l \right)} + 1 \right)}}{2} - \frac{\sqrt{\frac{1}{\left(\cos{\left(l \right)} - 1\right) \left(\cos{\left(l \right)} + 1\right)}} \log{\left(\sqrt{\frac{1}{\left(\cos{\left(l \right)} - 1\right) \left(\cos{\left(l \right)} + 1\right)}} \cos^{2}{\left(l \right)} - \sqrt{\frac{1}{\left(\cos{\left(l \right)} - 1\right) \left(\cos{\left(l \right)} + 1\right)}} - \cos{\left(l \right)} + 1 \right)}}{2} 
=
= 
    ____________________________    /      ____________________________                ____________________________        \       ____________________________    /        ____________________________                ____________________________        \       ____________________________    /    ____________________________                ____________________________        \       ____________________________    /        ____________________________                ____________________________        \
   /             1                  |     /             1                             /             1                  2   |      /             1                  |       /             1                             /             1                  2   |      /             1                  |   /             1                             /             1                  2   |      /             1                  |       /             1                             /             1                  2   |
  /  -------------------------- *log|-   /  --------------------------  - cos(l) +   /  -------------------------- *cos (l)|     /  -------------------------- *log|1 +   /  --------------------------  - cos(l) -   /  -------------------------- *cos (l)|     /  -------------------------- *log|  /  --------------------------  - cos(l) -   /  -------------------------- *cos (l)|     /  -------------------------- *log|1 -   /  --------------------------  - cos(l) +   /  -------------------------- *cos (l)|
\/   (1 + cos(l))*(-1 + cos(l))     \  \/   (1 + cos(l))*(-1 + cos(l))             \/   (1 + cos(l))*(-1 + cos(l))         /   \/   (1 + cos(l))*(-1 + cos(l))     \    \/   (1 + cos(l))*(-1 + cos(l))             \/   (1 + cos(l))*(-1 + cos(l))         /   \/   (1 + cos(l))*(-1 + cos(l))     \\/   (1 + cos(l))*(-1 + cos(l))             \/   (1 + cos(l))*(-1 + cos(l))         /   \/   (1 + cos(l))*(-1 + cos(l))     \    \/   (1 + cos(l))*(-1 + cos(l))             \/   (1 + cos(l))*(-1 + cos(l))         /
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                                                             2                                                                                                                               2                                                                                                                              2                                                                                                                              2
−1(cos⁡(l)−1)(cos⁡(l)+1)log⁡(−1(cos⁡(l)−1)(cos⁡(l)+1)cos⁡2(l)+1(cos⁡(l)−1)(cos⁡(l)+1)−cos⁡(l))2+1(cos⁡(l)−1)(cos⁡(l)+1)log⁡(1(cos⁡(l)−1)(cos⁡(l)+1)cos⁡2(l)−1(cos⁡(l)−1)(cos⁡(l)+1)−cos⁡(l))2+1(cos⁡(l)−1)(cos⁡(l)+1)log⁡(−1(cos⁡(l)−1)(cos⁡(l)+1)cos⁡2(l)+1(cos⁡(l)−1)(cos⁡(l)+1)−cos⁡(l)+1)2−1(cos⁡(l)−1)(cos⁡(l)+1)log⁡(1(cos⁡(l)−1)(cos⁡(l)+1)cos⁡2(l)−1(cos⁡(l)−1)(cos⁡(l)+1)−cos⁡(l)+1)2- \frac{\sqrt{\frac{1}{\left(\cos{\left(l \right)} - 1\right) \left(\cos{\left(l \right)} + 1\right)}} \log{\left(- \sqrt{\frac{1}{\left(\cos{\left(l \right)} - 1\right) \left(\cos{\left(l \right)} + 1\right)}} \cos^{2}{\left(l \right)} + \sqrt{\frac{1}{\left(\cos{\left(l \right)} - 1\right) \left(\cos{\left(l \right)} + 1\right)}} - \cos{\left(l \right)} \right)}}{2} + \frac{\sqrt{\frac{1}{\left(\cos{\left(l \right)} - 1\right) \left(\cos{\left(l \right)} + 1\right)}} \log{\left(\sqrt{\frac{1}{\left(\cos{\left(l \right)} - 1\right) \left(\cos{\left(l \right)} + 1\right)}} \cos^{2}{\left(l \right)} - \sqrt{\frac{1}{\left(\cos{\left(l \right)} - 1\right) \left(\cos{\left(l \right)} + 1\right)}} - \cos{\left(l \right)} \right)}}{2} + \frac{\sqrt{\frac{1}{\left(\cos{\left(l \right)} - 1\right) \left(\cos{\left(l \right)} + 1\right)}} \log{\left(- \sqrt{\frac{1}{\left(\cos{\left(l \right)} - 1\right) \left(\cos{\left(l \right)} + 1\right)}} \cos^{2}{\left(l \right)} + \sqrt{\frac{1}{\left(\cos{\left(l \right)} - 1\right) \left(\cos{\left(l \right)} + 1\right)}} - \cos{\left(l \right)} + 1 \right)}}{2} - \frac{\sqrt{\frac{1}{\left(\cos{\left(l \right)} - 1\right) \left(\cos{\left(l \right)} + 1\right)}} \log{\left(\sqrt{\frac{1}{\left(\cos{\left(l \right)} - 1\right) \left(\cos{\left(l \right)} + 1\right)}} \cos^{2}{\left(l \right)} - \sqrt{\frac{1}{\left(\cos{\left(l \right)} - 1\right) \left(\cos{\left(l \right)} + 1\right)}} - \cos{\left(l \right)} + 1 \right)}}{2} 
sqrt(1/((1 + cos(l))*(-1 + cos(l))))*log(-sqrt(1/((1 + cos(l))*(-1 + cos(l)))) - cos(l) + sqrt(1/((1 + cos(l))*(-1 + cos(l))))*cos(l)^2)/2 + sqrt(1/((1 + cos(l))*(-1 + cos(l))))*log(1 + sqrt(1/((1 + cos(l))*(-1 + cos(l)))) - cos(l) - sqrt(1/((1 + cos(l))*(-1 + cos(l))))*cos(l)^2)/2 - sqrt(1/((1 + cos(l))*(-1 + cos(l))))*log(sqrt(1/((1 + cos(l))*(-1 + cos(l)))) - cos(l) - sqrt(1/((1 + cos(l))*(-1 + cos(l))))*cos(l)^2)/2 - sqrt(1/((1 + cos(l))*(-1 + cos(l))))*log(1 - sqrt(1/((1 + cos(l))*(-1 + cos(l)))) - cos(l) + sqrt(1/((1 + cos(l))*(-1 + cos(l))))*cos(l)^2)/2

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

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