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Resolución de integrales/ cos^2/ sqrt2/ 1/cos/ 2cosx/ ((1/cos^2)x)-2cosx+sqrt2

Integral de ((1/cos^2)x)-2cosx+sqrt2 dx

Solución

Ha introducido [src]

  1                                
  /                                
 |                                 
 |  /   x                   ___\   
 |  |------- - 2*cos(x) + \/ 2 | dx
 |  |   2                      |   
 |  \cos (x)                   /   
 |                                 
/                                  
0

$$\int\limits_{0}^{1} \left(\left(\frac{x}{\cos^{2}{\left(x \right)}} - 2 \cos{\left(x \right)}\right) + \sqrt{2}\right)\, dx$$

Integral(x/cos(x)^2 - 2*cos(x) + sqrt(2), (x, 0, 1))

Solución detallada

Integramos término a término:
1. Integramos término a término:
  1. No puedo encontrar los pasos en la búsqueda de esta integral.
    
    Pero la integral
    
    $- \frac{2 x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1}$
  1. La integral del producto de una función por una constante es la constante por la integral de esta función:
    
    $\int \left(- 2 \cos{\left(x \right)}\right)\, dx = - 2 \int \cos{\left(x \right)}\, dx$
    1. La integral del coseno es seno:
      
      $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
    Por lo tanto, el resultado es: $- 2 \sin{\left(x \right)}$
  El resultado es: $- \frac{2 x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - 2 \sin{\left(x \right)} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1}$
1. La integral de las constantes tienen esta constante multiplicada por la variable de integración:
  
  $\int \sqrt{2}\, dx = \sqrt{2} x$
El resultado es: $\sqrt{2} x - \frac{2 x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - 2 \sin{\left(x \right)} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1}$
Ahora simplificar:

$\frac{\frac{\left(\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}\right) \left(\cos{\left(x \right)} - 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right)}{2} + \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \left(x \tan{\left(x \right)} + \sqrt{2} x - 2 \sin{\left(x \right)}\right) \cos{\left(x \right)} + \left(- \log{\left(\frac{2}{\cos{\left(x \right)} + 1} \right)} \tan^{2}{\left(\frac{x}{2} \right)} + \log{\left(\frac{2}{\cos{\left(x \right)} + 1} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}\right) \cos{\left(x \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \cos{\left(x \right)}}$
Añadimos la constante de integración:

$\frac{\frac{\left(\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}\right) \left(\cos{\left(x \right)} - 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right)}{2} + \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \left(x \tan{\left(x \right)} + \sqrt{2} x - 2 \sin{\left(x \right)}\right) \cos{\left(x \right)} + \left(- \log{\left(\frac{2}{\cos{\left(x \right)} + 1} \right)} \tan^{2}{\left(\frac{x}{2} \right)} + \log{\left(\frac{2}{\cos{\left(x \right)} + 1} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}\right) \cos{\left(x \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \cos{\left(x \right)}}+ \mathrm{constant}$

Respuesta:

$\frac{\frac{\left(\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}\right) \left(\cos{\left(x \right)} - 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right)}{2} + \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \left(x \tan{\left(x \right)} + \sqrt{2} x - 2 \sin{\left(x \right)}\right) \cos{\left(x \right)} + \left(- \log{\left(\frac{2}{\cos{\left(x \right)} + 1} \right)} \tan^{2}{\left(\frac{x}{2} \right)} + \log{\left(\frac{2}{\cos{\left(x \right)} + 1} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}\right) \cos{\left(x \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \cos{\left(x \right)}}+ \mathrm{constant}$

Respuesta (Indefinida) [src]

  /                                                              /       2/x\\      /       /x\\      /        /x\\      2/x\    /       /x\\      2/x\    /        /x\\      2/x\    /       2/x\\           /x\ 
 |                                                            log|1 + tan |-||   log|1 + tan|-||   log|-1 + tan|-||   tan |-|*log|1 + tan|-||   tan |-|*log|-1 + tan|-||   tan |-|*log|1 + tan |-||    2*x*tan|-| 
 | /   x                   ___\                         ___      \        \2//      \       \2//      \        \2//       \2/    \       \2//       \2/    \        \2//       \2/    \        \2//           \2/ 
 | |------- - 2*cos(x) + \/ 2 | dx = C - 2*sin(x) + x*\/ 2  + ---------------- - --------------- - ---------------- + ----------------------- + ------------------------ - ------------------------ - ------------
 | |   2                      |                                         2/x\               2/x\              2/x\                   2/x\                      2/x\                       2/x\                 2/x\
 | \cos (x)                   /                                 -1 + tan |-|       -1 + tan |-|      -1 + tan |-|           -1 + tan |-|              -1 + tan |-|               -1 + tan |-|         -1 + tan |-|
 |                                                                       \2/                \2/               \2/                    \2/                       \2/                        \2/                  \2/
/

$$\int \left(\left(\frac{x}{\cos^{2}{\left(x \right)}} - 2 \cos{\left(x \right)}\right) + \sqrt{2}\right)\, dx = C + \sqrt{2} x - \frac{2 x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - 2 \sin{\left(x \right)} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1}$$

Simplificar

Gráfica

Trazar el gráfico f(x)

Respuesta [src]

                      /       2     \                                                                             2                                      2                             2         /       2     \
  ___              log\1 + tan (1/2)/          pi*I + log(1 - tan(1/2))   log(1 + tan(1/2))     2*tan(1/2)     tan (1/2)*(pi*I + log(1 - tan(1/2)))   tan (1/2)*log(1 + tan(1/2))   tan (1/2)*log\1 + tan (1/2)/
\/ 2  - 2*sin(1) + ------------------ - pi*I - ------------------------ - ----------------- - -------------- + ------------------------------------ + --------------------------- - ----------------------------
                             2                              2                       2                 2                           2                                  2                             2            
                     -1 + tan (1/2)                 -1 + tan (1/2)          -1 + tan (1/2)    -1 + tan (1/2)              -1 + tan (1/2)                     -1 + tan (1/2)                -1 + tan (1/2)

$$- 2 \sin{\left(1 \right)} + \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} + \sqrt{2} - \frac{2 \tan{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - i \pi + \frac{\left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}}$$

                      /       2     \                                                                             2                                      2                             2         /       2     \
  ___              log\1 + tan (1/2)/          pi*I + log(1 - tan(1/2))   log(1 + tan(1/2))     2*tan(1/2)     tan (1/2)*(pi*I + log(1 - tan(1/2)))   tan (1/2)*log(1 + tan(1/2))   tan (1/2)*log\1 + tan (1/2)/
\/ 2  - 2*sin(1) + ------------------ - pi*I - ------------------------ - ----------------- - -------------- + ------------------------------------ + --------------------------- - ----------------------------
                             2                              2                       2                 2                           2                                  2                             2            
                     -1 + tan (1/2)                 -1 + tan (1/2)          -1 + tan (1/2)    -1 + tan (1/2)              -1 + tan (1/2)                     -1 + tan (1/2)                -1 + tan (1/2)

sqrt(2) - 2*sin(1) + log(1 + tan(1/2)^2)/(-1 + tan(1/2)^2) - pi*i - (pi*i + log(1 - tan(1/2)))/(-1 + tan(1/2)^2) - log(1 + tan(1/2))/(-1 + tan(1/2)^2) - 2*tan(1/2)/(-1 + tan(1/2)^2) + tan(1/2)^2*(pi*i + log(1 - tan(1/2)))/(-1 + tan(1/2)^2) + tan(1/2)^2*log(1 + tan(1/2))/(-1 + tan(1/2)^2) - tan(1/2)^2*log(1 + tan(1/2)^2)/(-1 + tan(1/2)^2)

Respuesta numérica [src]

0.67305284702619

0.67305284702619

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

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Expresiones con funciones

Integral de ((1/cos^2)x)-2cosx+sqrt2 dx

Límites de integración:

Gráfico:

Definida a trozos:

Solución