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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                   
  /                   
 |                    
 |      2             
 |   sin (x)*cos(x)   
 |  --------------- dx
 |  sin(x) + cos(x)   
 |                    
/                     
0
$$\int\limits_{0}^{1} \frac{\sin^{2}{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}}\, dx$$ 
Integral((sin(x)^2*cos(x))/(sin(x) + cos(x)), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                                                                                                                                                                                                                                                                                           
 |                             /       2/x\        /x\\           /       2/x\\                     /x\                        3/x\                        2/x\              4/x\    /       2/x\        /x\\       4/x\    /       2/x\\        2/x\    /       2/x\\        2/x\    /       2/x\        /x\\
 |     2                    log|1 - tan |-| + 2*tan|-||        log|1 + tan |-||                2*tan|-|                   2*tan |-|                   4*tan |-|           tan |-|*log|1 - tan |-| + 2*tan|-||    tan |-|*log|1 + tan |-||   2*tan |-|*log|1 + tan |-||   2*tan |-|*log|1 - tan |-| + 2*tan|-||
 |  sin (x)*cos(x)             \        \2/        \2//           \        \2//                     \2/                         \2/                         \2/               \2/    \        \2/        \2//        \2/    \        \2//         \2/    \        \2//         \2/    \        \2/        \2//
 | --------------- dx = C + --------------------------- - ------------------------- - ------------------------- + ------------------------- + ------------------------- + ----------------------------------- - ------------------------- - -------------------------- + -------------------------------------
 | sin(x) + cos(x)                    4/x\        2/x\             4/x\        2/x\            4/x\        2/x\            4/x\        2/x\            4/x\        2/x\                 4/x\        2/x\                 4/x\        2/x\            4/x\        2/x\                   4/x\        2/x\      
 |                           4 + 4*tan |-| + 8*tan |-|    4 + 4*tan |-| + 8*tan |-|   4 + 4*tan |-| + 8*tan |-|   4 + 4*tan |-| + 8*tan |-|   4 + 4*tan |-| + 8*tan |-|        4 + 4*tan |-| + 8*tan |-|        4 + 4*tan |-| + 8*tan |-|   4 + 4*tan |-| + 8*tan |-|          4 + 4*tan |-| + 8*tan |-|      
/                                      \2/         \2/              \2/         \2/             \2/         \2/             \2/         \2/             \2/         \2/                  \2/         \2/                  \2/         \2/             \2/         \2/                    \2/         \2/
$$\int \frac{\sin^{2}{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}}\, dx = C - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} - \frac{2 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{\log{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{2 \log{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{\log{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 1 \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{2 \tan^{3}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{4 \tan^{2}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} - \frac{2 \tan{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
   /       2                  \            /       2     \                                                      3                               2                    4         /       2                  \       4         /       2     \        2         /       2     \        2         /       2                  \
log\1 - tan (1/2) + 2*tan(1/2)/         log\1 + tan (1/2)/                  2*tan(1/2)                     2*tan (1/2)                     4*tan (1/2)            tan (1/2)*log\1 - tan (1/2) + 2*tan(1/2)/    tan (1/2)*log\1 + tan (1/2)/   2*tan (1/2)*log\1 + tan (1/2)/   2*tan (1/2)*log\1 - tan (1/2) + 2*tan(1/2)/
------------------------------- - ----------------------------- - ----------------------------- + ----------------------------- + ----------------------------- + ----------------------------------------- - ----------------------------- - ------------------------------ + -------------------------------------------
          4             2                  4             2                 4             2                 4             2                 4             2                       4             2                       4             2                 4             2                         4             2            
 4 + 4*tan (1/2) + 8*tan (1/2)    4 + 4*tan (1/2) + 8*tan (1/2)   4 + 4*tan (1/2) + 8*tan (1/2)   4 + 4*tan (1/2) + 8*tan (1/2)   4 + 4*tan (1/2) + 8*tan (1/2)         4 + 4*tan (1/2) + 8*tan (1/2)         4 + 4*tan (1/2) + 8*tan (1/2)   4 + 4*tan (1/2) + 8*tan (1/2)           4 + 4*tan (1/2) + 8*tan (1/2)
$$- \frac{2 \tan{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{2 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{\log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + 1 + 2 \tan{\left(\frac{1}{2} \right)} \right)} \tan^{4}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{2 \tan^{3}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{2 \log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + 1 + 2 \tan{\left(\frac{1}{2} \right)} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{\log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + 1 + 2 \tan{\left(\frac{1}{2} \right)} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{4 \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4}$$ 
=
= 
   /       2                  \            /       2     \                                                      3                               2                    4         /       2                  \       4         /       2     \        2         /       2     \        2         /       2                  \
log\1 - tan (1/2) + 2*tan(1/2)/         log\1 + tan (1/2)/                  2*tan(1/2)                     2*tan (1/2)                     4*tan (1/2)            tan (1/2)*log\1 - tan (1/2) + 2*tan(1/2)/    tan (1/2)*log\1 + tan (1/2)/   2*tan (1/2)*log\1 + tan (1/2)/   2*tan (1/2)*log\1 - tan (1/2) + 2*tan(1/2)/
------------------------------- - ----------------------------- - ----------------------------- + ----------------------------- + ----------------------------- + ----------------------------------------- - ----------------------------- - ------------------------------ + -------------------------------------------
          4             2                  4             2                 4             2                 4             2                 4             2                       4             2                       4             2                 4             2                         4             2            
 4 + 4*tan (1/2) + 8*tan (1/2)    4 + 4*tan (1/2) + 8*tan (1/2)   4 + 4*tan (1/2) + 8*tan (1/2)   4 + 4*tan (1/2) + 8*tan (1/2)   4 + 4*tan (1/2) + 8*tan (1/2)         4 + 4*tan (1/2) + 8*tan (1/2)         4 + 4*tan (1/2) + 8*tan (1/2)   4 + 4*tan (1/2) + 8*tan (1/2)           4 + 4*tan (1/2) + 8*tan (1/2)
$$- \frac{2 \tan{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{2 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{\log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + 1 + 2 \tan{\left(\frac{1}{2} \right)} \right)} \tan^{4}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{2 \tan^{3}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{2 \log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + 1 + 2 \tan{\left(\frac{1}{2} \right)} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{\log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + 1 + 2 \tan{\left(\frac{1}{2} \right)} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{4 \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4}$$ 
log(1 - tan(1/2)^2 + 2*tan(1/2))/(4 + 4*tan(1/2)^4 + 8*tan(1/2)^2) - log(1 + tan(1/2)^2)/(4 + 4*tan(1/2)^4 + 8*tan(1/2)^2) - 2*tan(1/2)/(4 + 4*tan(1/2)^4 + 8*tan(1/2)^2) + 2*tan(1/2)^3/(4 + 4*tan(1/2)^4 + 8*tan(1/2)^2) + 4*tan(1/2)^2/(4 + 4*tan(1/2)^4 + 8*tan(1/2)^2) + tan(1/2)^4*log(1 - tan(1/2)^2 + 2*tan(1/2))/(4 + 4*tan(1/2)^4 + 8*tan(1/2)^2) - tan(1/2)^4*log(1 + tan(1/2)^2)/(4 + 4*tan(1/2)^4 + 8*tan(1/2)^2) - 2*tan(1/2)^2*log(1 + tan(1/2)^2)/(4 + 4*tan(1/2)^4 + 8*tan(1/2)^2) + 2*tan(1/2)^2*log(1 - tan(1/2)^2 + 2*tan(1/2))/(4 + 4*tan(1/2)^4 + 8*tan(1/2)^2)
Respuesta numérica [src] 
0.144198093094028
0.144198093094028

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

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