Integral de cos(x)*ln(cos(x)) dx

Solución

Ha introducido [src]

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0

\int\limits_{0}^{1} \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}\, dx

Integral(cos(x)*log(cos(x)), (x, 0, 1))

Solución detallada

Usamos la integración por partes:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

que $u{\left(x \right)} = \log{\left(\cos{\left(x \right)} \right)}$ y que $\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}$ .

Entonces $\operatorname{du}{\left(x \right)} = - \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$ .

Para buscar $v{\left(x \right)}$ :
1. La integral del coseno es seno:
  
  $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
Ahora resolvemos podintegral.
La integral del producto de una función por una constante es la constante por la integral de esta función:

$\int \left(- \frac{\sin^{2}{\left(x \right)}}{\cos{\left(x \right)}}\right)\, dx = - \int \frac{\sin^{2}{\left(x \right)}}{\cos{\left(x \right)}}\, dx$
1. No puedo encontrar los pasos en la búsqueda de esta integral.
  
  Pero la integral
  
  $- \frac{\log{\left(\sin{\left(x \right)} - 1 \right)}}{2} + \frac{\log{\left(\sin{\left(x \right)} + 1 \right)}}{2} - \sin{\left(x \right)}$
Por lo tanto, el resultado es: $\frac{\log{\left(\sin{\left(x \right)} - 1 \right)}}{2} - \frac{\log{\left(\sin{\left(x \right)} + 1 \right)}}{2} + \sin{\left(x \right)}$
Añadimos la constante de integración:

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

Respuesta:

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

Respuesta (Indefinida) [src]

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

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

Simplificar

Gráfica

Trazar el gráfico f(x)

Respuesta [src]

                          /                     2       \                                                                                     /                     2       \                                        /                     2       \         
                          |      1           tan (1/2)  |                                                                           2         |      1           tan (1/2)  |                                        |      1           tan (1/2)  |         
                       log|------------- - -------------|                                                                        tan (1/2)*log|------------- - -------------|                                   2*log|------------- - -------------|*tan(1/2)
     /       2     \      |       2               2     |                                            2         /       2     \                |       2               2     |        2                               |       2               2     |         
  log\1 + tan (1/2)/      \1 + tan (1/2)   1 + tan (1/2)/     2*tan(1/2)    2*log(1 + tan(1/2))   tan (1/2)*log\1 + tan (1/2)/                \1 + tan (1/2)   1 + tan (1/2)/   2*tan (1/2)*log(1 + tan(1/2))        \1 + tan (1/2)   1 + tan (1/2)/         
- ------------------ - ---------------------------------- - ------------- + ------------------- - ---------------------------- - -------------------------------------------- + ----------------------------- + ---------------------------------------------
           2                            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)                           1 + tan (1/2)

- \frac{2 \tan{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \log{\left(- \frac{\tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{1}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} \right)} \tan{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(- \frac{\tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{1}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(- \frac{\tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{1}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1}

                          /                     2       \                                                                                     /                     2       \                                        /                     2       \         
                          |      1           tan (1/2)  |                                                                           2         |      1           tan (1/2)  |                                        |      1           tan (1/2)  |         
                       log|------------- - -------------|                                                                        tan (1/2)*log|------------- - -------------|                                   2*log|------------- - -------------|*tan(1/2)
     /       2     \      |       2               2     |                                            2         /       2     \                |       2               2     |        2                               |       2               2     |         
  log\1 + tan (1/2)/      \1 + tan (1/2)   1 + tan (1/2)/     2*tan(1/2)    2*log(1 + tan(1/2))   tan (1/2)*log\1 + tan (1/2)/                \1 + tan (1/2)   1 + tan (1/2)/   2*tan (1/2)*log(1 + tan(1/2))        \1 + tan (1/2)   1 + tan (1/2)/         
- ------------------ - ---------------------------------- - ------------- + ------------------- - ---------------------------- - -------------------------------------------- + ----------------------------- + ---------------------------------------------
           2                            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)                           1 + tan (1/2)

- \frac{2 \tan{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \log{\left(- \frac{\tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{1}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} \right)} \tan{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(- \frac{\tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{1}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(- \frac{\tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{1}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1}

-log(1 + tan(1/2)^2)/(1 + tan(1/2)^2) - log(1/(1 + tan(1/2)^2) - tan(1/2)^2/(1 + tan(1/2)^2))/(1 + tan(1/2)^2) - 2*tan(1/2)/(1 + tan(1/2)^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) - tan(1/2)^2*log(1/(1 + tan(1/2)^2) - tan(1/2)^2/(1 + tan(1/2)^2))/(1 + tan(1/2)^2) + 2*tan(1/2)^2*log(1 + tan(1/2))/(1 + tan(1/2)^2) + 2*log(1/(1 + tan(1/2)^2) - tan(1/2)^2/(1 + tan(1/2)^2))*tan(1/2)/(1 + tan(1/2)^2)

Respuesta numérica [src]

-0.133311626233908

-0.133311626233908

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Integral de cos(x)*ln(cos(x)) dx

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Solución