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Resolución de integrales/ cos^2/ sin(x)/ (x*sin(x))/cos^2(x)

Integral de (x*sin(x))/cos^2(x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1            
  /            
 |             
 |  x*sin(x)   
 |  -------- dx
 |     2       
 |  cos (x)    
 |             
/              
0
$$\int\limits_{0}^{1} \frac{x \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}\, dx$$ 
Integral((x*sin(x))/cos(x)^2, (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                     /       /x\\                     /        /x\\      2/x\    /        /x\\         2/x\        2/x\    /       /x\\
 |                   log|1 + tan|-||                  log|-1 + tan|-||   tan |-|*log|-1 + tan|-||    x*tan |-|     tan |-|*log|1 + tan|-||
 | x*sin(x)             \       \2//        x            \        \2//       \2/    \        \2//          \2/         \2/    \       \2//
 | -------- dx = C + --------------- - ------------ - ---------------- + ------------------------ - ------------ - -----------------------
 |    2                        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 |-|     
 |                              \2/             \2/              \2/                    \2/                  \2/                  \2/     
/
$$\int \frac{x \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}\, dx = C - \frac{x \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{x}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
                                                   2                                         2                                      2                       
        1          log(1 + tan(1/2))            tan (1/2)      pi*I + log(1 - tan(1/2))   tan (1/2)*(pi*I + log(1 - tan(1/2)))   tan (1/2)*log(1 + tan(1/2))
- -------------- + ----------------- - pi*I - -------------- - ------------------------ + ------------------------------------ - ---------------------------
          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)
$$\frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{1}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - i \pi + \frac{\left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}}$$ 
=
= 
                                                   2                                         2                                      2                       
        1          log(1 + tan(1/2))            tan (1/2)      pi*I + log(1 - tan(1/2))   tan (1/2)*(pi*I + log(1 - tan(1/2)))   tan (1/2)*log(1 + tan(1/2))
- -------------- + ----------------- - pi*I - -------------- - ------------------------ + ------------------------------------ - ---------------------------
          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)
$$\frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{1}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - i \pi + \frac{\left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}}$$ 
-1/(-1 + tan(1/2)^2) + log(1 + tan(1/2))/(-1 + tan(1/2)^2) - pi*i - tan(1/2)^2/(-1 + tan(1/2)^2) - (pi*i + log(1 - 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)
Respuesta numérica [src] 
0.624624546797409
0.624624546797409

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

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