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Resolución de integrales/ cos^2/ -sinx/ (1/(cos^2)x)-sinx

Integral de (1/(cos^2)x)-sinx dx

Solución

Ha introducido [src]

  pi                      
  --                      
  4                       
   /                      
  |                       
  |  /   x            \   
  |  |------- - sin(x)| dx
  |  |   2            |   
  |  \cos (x)         /   
  |                       
 /                        
-pi                       
----                      
 4

$$\int\limits_{- \frac{\pi}{4}}^{\frac{\pi}{4}} \left(\frac{x}{\cos^{2}{\left(x \right)}} - \sin{\left(x \right)}\right)\, dx$$

Integral(x/cos(x)^2 - sin(x), (x, -pi/4, pi/4))

Solución detallada

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:
  
  $\int \left(- \sin{\left(x \right)}\right)\, dx = - \int \sin{\left(x \right)}\, dx$
  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: $\cos{\left(x \right)}$
El resultado es: $- \frac{2 x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \cos{\left(x \right)} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1}$
Ahora simplificar:

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

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

Respuesta:

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

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/          
 | |------- - sin(x)| dx = C + ---------------- - --------------- - ---------------- + ----------------------- + ------------------------ - ------------------------ - ------------ + cos(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/         
/

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

Simplificar

Gráfica

Trazar el gráfico f(x)

Respuesta [src]

                                           /                2\      /               2\                                                           2    /               2\               2                                       2                         2                                  2                              2    /                2\                                                 
          /  ___\        /      ___\       |    /       ___\ |      |    /      ___\ |             /      ___\          /  ___\       /      ___\     |    /      ___\ |   /       ___\  /          /      ___\\   /       ___\     /  ___\   /      ___\  /          /  ___\\   /      ___\     /      ___\   /       ___\     |    /       ___\ |          /      ___\            /       ___\    
pi*I + log\\/ 2 /     log\2 - \/ 2 /    log\1 + \-1 + \/ 2 / /   log\1 + \1 - \/ 2 / /   pi*I + log\2 - \/ 2 /       log\\/ 2 /       \1 - \/ 2 / *log\1 + \1 - \/ 2 / /   \-1 + \/ 2 / *\pi*I + log\2 - \/ 2 //   \-1 + \/ 2 / *log\\/ 2 /   \1 - \/ 2 / *\pi*I + log\\/ 2 //   \1 - \/ 2 / *log\2 - \/ 2 /   \-1 + \/ 2 / *log\1 + \-1 + \/ 2 / /       pi*\1 - \/ 2 /         pi*\-1 + \/ 2 /    
----------------- + ----------------- + ---------------------- - --------------------- - --------------------- - ------------------ + ---------------------------------- + ------------------------------------- + ------------------------ - -------------------------------- - --------------------------- - ------------------------------------ - --------------------- - ----------------------
                2                   2                      2                       2                        2                     2                           2                                       2                                2                             2                                2                                  2              /                2\     /                 2\
     /      ___\         /      ___\           /       ___\             /      ___\             /       ___\          /       ___\                 /      ___\                            /       ___\                     /       ___\                   /      ___\                      /      ___\                       /       ___\               |     /      ___\ |     |     /       ___\ |
-1 + \1 - \/ 2 /    -1 + \1 - \/ 2 /      -1 + \-1 + \/ 2 /        -1 + \1 - \/ 2 /        -1 + \-1 + \/ 2 /     -1 + \-1 + \/ 2 /            -1 + \1 - \/ 2 /                       -1 + \-1 + \/ 2 /                -1 + \-1 + \/ 2 /              -1 + \1 - \/ 2 /                 -1 + \1 - \/ 2 /                  -1 + \-1 + \/ 2 /             2*\-1 + \1 - \/ 2 / /   2*\-1 + \-1 + \/ 2 / /

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

                                           /                2\      /               2\                                                           2    /               2\               2                                       2                         2                                  2                              2    /                2\                                                 
          /  ___\        /      ___\       |    /       ___\ |      |    /      ___\ |             /      ___\          /  ___\       /      ___\     |    /      ___\ |   /       ___\  /          /      ___\\   /       ___\     /  ___\   /      ___\  /          /  ___\\   /      ___\     /      ___\   /       ___\     |    /       ___\ |          /      ___\            /       ___\    
pi*I + log\\/ 2 /     log\2 - \/ 2 /    log\1 + \-1 + \/ 2 / /   log\1 + \1 - \/ 2 / /   pi*I + log\2 - \/ 2 /       log\\/ 2 /       \1 - \/ 2 / *log\1 + \1 - \/ 2 / /   \-1 + \/ 2 / *\pi*I + log\2 - \/ 2 //   \-1 + \/ 2 / *log\\/ 2 /   \1 - \/ 2 / *\pi*I + log\\/ 2 //   \1 - \/ 2 / *log\2 - \/ 2 /   \-1 + \/ 2 / *log\1 + \-1 + \/ 2 / /       pi*\1 - \/ 2 /         pi*\-1 + \/ 2 /    
----------------- + ----------------- + ---------------------- - --------------------- - --------------------- - ------------------ + ---------------------------------- + ------------------------------------- + ------------------------ - -------------------------------- - --------------------------- - ------------------------------------ - --------------------- - ----------------------
                2                   2                      2                       2                        2                     2                           2                                       2                                2                             2                                2                                  2              /                2\     /                 2\
     /      ___\         /      ___\           /       ___\             /      ___\             /       ___\          /       ___\                 /      ___\                            /       ___\                     /       ___\                   /      ___\                      /      ___\                       /       ___\               |     /      ___\ |     |     /       ___\ |
-1 + \1 - \/ 2 /    -1 + \1 - \/ 2 /      -1 + \-1 + \/ 2 /        -1 + \1 - \/ 2 /        -1 + \-1 + \/ 2 /     -1 + \-1 + \/ 2 /            -1 + \1 - \/ 2 /                       -1 + \-1 + \/ 2 /                -1 + \-1 + \/ 2 /              -1 + \1 - \/ 2 /                 -1 + \1 - \/ 2 /                  -1 + \-1 + \/ 2 /             2*\-1 + \1 - \/ 2 / /   2*\-1 + \-1 + \/ 2 / /

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

Respuesta numérica [src]

0.0

0.0

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

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Expresiones idénticas

Expresiones semejantes

Expresiones con funciones

Integral de (1/(cos^2)x)-sinx dx

Límites de integración:

Gráfico:

Definida a trozos:

Solución