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Expresiones con funciones
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Resolución de integrales/ cos^2/ 4sinx/ (4sinx+(8x^3)-11/(cos^2)x)

Integral de (4sinx+(8x^3)-11/(cos^2)x) dx

Solución

Ha introducido [src]

  1                                 
  /                                 
 |                                  
 |  /              3      11    \   
 |  |4*sin(x) + 8*x  - -------*x| dx
 |  |                     2     |   
 |  \                  cos (x)  /   
 |                                  
/                                   
0

$$\int\limits_{0}^{1} \left(- x \frac{11}{\cos^{2}{\left(x \right)}} + \left(8 x^{3} + 4 \sin{\left(x \right)}\right)\right)\, dx$$

Integral(4*sin(x) + 8*x^3 - 11/cos(x)^2*x, (x, 0, 1))

Solución detallada

Integramos término a término:
1. La integral del producto de una función por una constante es la constante por la integral de esta función:
  
  $\int \left(- x \frac{11}{\cos^{2}{\left(x \right)}}\right)\, dx = - \int \frac{11 x}{\cos^{2}{\left(x \right)}}\, dx$
  1. La integral del producto de una función por una constante es la constante por la integral de esta función:
    
    $\int \frac{11 x}{\cos^{2}{\left(x \right)}}\, dx = 11 \int \frac{x}{\cos^{2}{\left(x \right)}}\, dx$
    1. No puedo encontrar los pasos en la búsqueda de esta integral.
      
      Pero la integral
      
      $- \frac{2 x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1}$
    Por lo tanto, el resultado es: $- \frac{22 x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{11 \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{11 \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{11 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{11 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{11 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{11 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1}$
  Por lo tanto, el resultado es: $\frac{22 x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{11 \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{11 \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{11 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{11 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{11 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{11 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1}$
1. Integramos término a término:
  1. La integral del producto de una función por una constante es la constante por la integral de esta función:
    
    $\int 8 x^{3}\, dx = 8 \int x^{3}\, dx$
    1. Integral $x^{n}$ es $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{3}\, dx = \frac{x^{4}}{4}$
    Por lo tanto, el resultado es: $2 x^{4}$
  1. La integral del producto de una función por una constante es la constante por la integral de esta función:
    
    $\int 4 \sin{\left(x \right)}\, dx = 4 \int \sin{\left(x \right)}\, dx$
    1. La integral del seno es un coseno menos:
      
      $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
    Por lo tanto, el resultado es: $- 4 \cos{\left(x \right)}$
  El resultado es: $2 x^{4} - 4 \cos{\left(x \right)}$
El resultado es: $2 x^{4} + \frac{22 x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - 4 \cos{\left(x \right)} - \frac{11 \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{11 \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{11 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{11 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{11 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{11 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1}$
Ahora simplificar:

$\frac{- \frac{11 \left(\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}\right) \left(\cos{\left(x \right)} - 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right)}{2} + \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \left(2 x^{4} - 11 x \tan{\left(x \right)} - 4 \cos{\left(x \right)}\right) \cos{\left(x \right)} + 11 \left(\log{\left(\frac{2}{\cos{\left(x \right)} + 1} \right)} \tan^{2}{\left(\frac{x}{2} \right)} - \log{\left(\frac{2}{\cos{\left(x \right)} + 1} \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}\right) \cos{\left(x \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \cos{\left(x \right)}}$
Añadimos la constante de integración:

$\frac{- \frac{11 \left(\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}\right) \left(\cos{\left(x \right)} - 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right)}{2} + \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \left(2 x^{4} - 11 x \tan{\left(x \right)} - 4 \cos{\left(x \right)}\right) \cos{\left(x \right)} + 11 \left(\log{\left(\frac{2}{\cos{\left(x \right)} + 1} \right)} \tan^{2}{\left(\frac{x}{2} \right)} - \log{\left(\frac{2}{\cos{\left(x \right)} + 1} \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}\right) \cos{\left(x \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \cos{\left(x \right)}}+ \mathrm{constant}$

Respuesta:

$\frac{- \frac{11 \left(\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}\right) \left(\cos{\left(x \right)} - 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right)}{2} + \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \left(2 x^{4} - 11 x \tan{\left(x \right)} - 4 \cos{\left(x \right)}\right) \cos{\left(x \right)} + 11 \left(\log{\left(\frac{2}{\cos{\left(x \right)} + 1} \right)} \tan^{2}{\left(\frac{x}{2} \right)} - \log{\left(\frac{2}{\cos{\left(x \right)} + 1} \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}\right) \cos{\left(x \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \cos{\left(x \right)}}+ \mathrm{constant}$

Respuesta (Indefinida) [src]

  /                                                               /       2/x\\         /       /x\\         /        /x\\         2/x\    /       /x\\         2/x\    /        /x\\         2/x\    /       2/x\\           /x\ 
 |                                                          11*log|1 + tan |-||   11*log|1 + tan|-||   11*log|-1 + tan|-||   11*tan |-|*log|1 + tan|-||   11*tan |-|*log|-1 + tan|-||   11*tan |-|*log|1 + tan |-||   22*x*tan|-| 
 | /              3      11    \                        4         \        \2//         \       \2//         \        \2//          \2/    \       \2//          \2/    \        \2//          \2/    \        \2//           \2/ 
 | |4*sin(x) + 8*x  - -------*x| dx = C - 4*cos(x) + 2*x  - ------------------- + ------------------ + ------------------- - -------------------------- - --------------------------- + --------------------------- + ------------
 | |                     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/
/

$$\int \left(- x \frac{11}{\cos^{2}{\left(x \right)}} + \left(8 x^{3} + 4 \sin{\left(x \right)}\right)\right)\, dx = C + 2 x^{4} + \frac{22 x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - 4 \cos{\left(x \right)} - \frac{11 \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{11 \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{11 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{11 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{11 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{11 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1}$$

Simplificar

Gráfica

Trazar el gráfico f(x)

Respuesta [src]

                     /       2     \                                                                                           2                                         2                                2         /       2     \
               11*log\1 + tan (1/2)/             11*(pi*I + log(1 - tan(1/2)))   11*log(1 + tan(1/2))    22*tan(1/2)     11*tan (1/2)*(pi*I + log(1 - tan(1/2)))   11*tan (1/2)*log(1 + tan(1/2))   11*tan (1/2)*log\1 + tan (1/2)/
6 - 4*cos(1) - --------------------- + 11*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)

$$\frac{22 \tan{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{11 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - 4 \cos{\left(1 \right)} + \frac{11 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{11 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{11 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} + 6 + \frac{11 \left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right)}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{11 \left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} + 11 i \pi$$

                     /       2     \                                                                                           2                                         2                                2         /       2     \
               11*log\1 + tan (1/2)/             11*(pi*I + log(1 - tan(1/2)))   11*log(1 + tan(1/2))    22*tan(1/2)     11*tan (1/2)*(pi*I + log(1 - tan(1/2)))   11*tan (1/2)*log(1 + tan(1/2))   11*tan (1/2)*log\1 + tan (1/2)/
6 - 4*cos(1) - --------------------- + 11*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)

6 - 4*cos(1) - 11*log(1 + tan(1/2)^2)/(-1 + tan(1/2)^2) + 11*pi*i + 11*(pi*i + log(1 - tan(1/2)))/(-1 + tan(1/2)^2) + 11*log(1 + tan(1/2))/(-1 + tan(1/2)^2) + 22*tan(1/2)/(-1 + tan(1/2)^2) - 11*tan(1/2)^2*(pi*i + log(1 - tan(1/2)))/(-1 + tan(1/2)^2) - 11*tan(1/2)^2*log(1 + tan(1/2))/(-1 + tan(1/2)^2) + 11*tan(1/2)^2*log(1 + tan(1/2)^2)/(-1 + tan(1/2)^2)

Respuesta numérica [src]

-6.52080302043033

-6.52080302043033

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

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Integral de (4sinx+(8x^3)-11/(cos^2)x) dx

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Definida a trozos:

Solución