Sr Examen

Integral de cosx*ln(cosx) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
  1                      
  /                      
 |                       
 |  cos(x)*log(cos(x)) dx
 |                       
/                        
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
  1. Usamos la integración por partes:

    que y que .

    Entonces .

    Para buscar :

    1. La integral del coseno es seno:

    Ahora resolvemos podintegral.

  2. La integral del producto de una función por una constante es la constante por la integral de esta función:

    1. No puedo encontrar los pasos en la búsqueda de esta integral.

      Pero la integral

    Por lo tanto, el resultado es:

  3. Añadimos la constante de integración:


Respuesta:

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

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