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Resolución de integrales/ (2sin(x)^3)/((cos(x)+1)^3)-(sin(x))/(cos(x)+1)+(3cos(x)*sin(x))/((cos(x)+1)^2)

Integral de (2sin(x)^3)/((cos(x)+1)^3)-(sin(x))/(cos(x)+1)+(3cos(x)*sin(x))/((cos(x)+1)^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                                                  
  /                                                  
 |                                                   
 |  /       3                                    \   
 |  |  2*sin (x)       sin(x)     3*cos(x)*sin(x)|   
 |  |------------- - ---------- + ---------------| dx
 |  |            3   cos(x) + 1                2 |   
 |  \(cos(x) + 1)                  (cos(x) + 1)  /   
 |                                                   
/                                                    
0
$$\int\limits_{0}^{1} \left(\frac{\sin{\left(x \right)} 3 \cos{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} + \left(\frac{2 \sin^{3}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{3}} - \frac{\sin{\left(x \right)}}{\cos{\left(x \right)} + 1}\right)\right)\, dx$$ 
Integral((2*sin(x)^3)/(cos(x) + 1)^3 - sin(x)/(cos(x) + 1) + ((3*cos(x))*sin(x))/(cos(x) + 1)^2, (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                                                                                                                                                                                                                                                                       
 |                                                                                                                                                                                                                                                                                        
 | /       3                                    \                                                       /       2              \               2                                                                         2                                                                
 | |  2*sin (x)       sin(x)     3*cos(x)*sin(x)|              3                   4               3*log\1 + cos (x) + 2*cos(x)/          2*sin (x)                   4*cos(x)              4*log(1 + cos(x))       4*cos (x)*log(1 + cos(x))   8*cos(x)*log(1 + cos(x))                  
 | |------------- - ---------- + ---------------| dx = C - ---------- + ------------------------ - ----------------------------- + ------------------------ + ------------------------ + ------------------------ + ------------------------- + ------------------------ + log(cos(x) + 1)
 | |            3   cos(x) + 1                2 |          1 + cos(x)            2                               2                          2                          2                          2                           2                          2                                
 | \(cos(x) + 1)                  (cos(x) + 1)  /                       2 + 2*cos (x) + 4*cos(x)                                   2 + 2*cos (x) + 4*cos(x)   2 + 2*cos (x) + 4*cos(x)   2 + 2*cos (x) + 4*cos(x)    2 + 2*cos (x) + 4*cos(x)   2 + 2*cos (x) + 4*cos(x)                  
 |                                                                                                                                                                                                                                                                                        
/
$$\int \left(\frac{\sin{\left(x \right)} 3 \cos{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} + \left(\frac{2 \sin^{3}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{3}} - \frac{\sin{\left(x \right)}}{\cos{\left(x \right)} + 1}\right)\right)\, dx = C + \log{\left(\cos{\left(x \right)} + 1 \right)} - \frac{3 \log{\left(\cos^{2}{\left(x \right)} + 2 \cos{\left(x \right)} + 1 \right)}}{2} + \frac{4 \log{\left(\cos{\left(x \right)} + 1 \right)} \cos^{2}{\left(x \right)}}{2 \cos^{2}{\left(x \right)} + 4 \cos{\left(x \right)} + 2} + \frac{8 \log{\left(\cos{\left(x \right)} + 1 \right)} \cos{\left(x \right)}}{2 \cos^{2}{\left(x \right)} + 4 \cos{\left(x \right)} + 2} + \frac{4 \log{\left(\cos{\left(x \right)} + 1 \right)}}{2 \cos^{2}{\left(x \right)} + 4 \cos{\left(x \right)} + 2} + \frac{2 \sin^{2}{\left(x \right)}}{2 \cos^{2}{\left(x \right)} + 4 \cos{\left(x \right)} + 2} + \frac{4 \cos{\left(x \right)}}{2 \cos^{2}{\left(x \right)} + 4 \cos{\left(x \right)} + 2} + \frac{4}{2 \cos^{2}{\left(x \right)} + 4 \cos{\left(x \right)} + 2} - \frac{3}{\cos{\left(x \right)} + 1}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
                                                                            2                                                                                                    2                                                                
1       3                   4               3*log(1 + cos(1))          2*sin (1)                   4*cos(1)              4*log(1 + cos(1))       3*cos(1)*log(1 + cos(1))   4*cos (1)*log(1 + cos(1))   8*cos(1)*log(1 + cos(1))                  
- - ---------- + ------------------------ - ----------------- + ------------------------ + ------------------------ + ------------------------ - ------------------------ + ------------------------- + ------------------------ + log(1 + cos(1))
2   1 + cos(1)            2                     1 + cos(1)               2                          2                          2                        1 + cos(1)                    2                          2                                
                 2 + 2*cos (1) + 4*cos(1)                       2 + 2*cos (1) + 4*cos(1)   2 + 2*cos (1) + 4*cos(1)   2 + 2*cos (1) + 4*cos(1)                               2 + 2*cos (1) + 4*cos(1)   2 + 2*cos (1) + 4*cos(1)
$$- \frac{3}{\cos{\left(1 \right)} + 1} - \frac{3 \log{\left(\cos{\left(1 \right)} + 1 \right)}}{\cos{\left(1 \right)} + 1} - \frac{3 \log{\left(\cos{\left(1 \right)} + 1 \right)} \cos{\left(1 \right)}}{\cos{\left(1 \right)} + 1} + \frac{4 \log{\left(\cos{\left(1 \right)} + 1 \right)} \cos^{2}{\left(1 \right)}}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} + \frac{2 \sin^{2}{\left(1 \right)}}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} + \frac{4 \log{\left(\cos{\left(1 \right)} + 1 \right)}}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} + \frac{8 \log{\left(\cos{\left(1 \right)} + 1 \right)} \cos{\left(1 \right)}}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} + \log{\left(\cos{\left(1 \right)} + 1 \right)} + \frac{4 \cos{\left(1 \right)}}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} + \frac{1}{2} + \frac{4}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}}$$ 
=
= 
                                                                            2                                                                                                    2                                                                
1       3                   4               3*log(1 + cos(1))          2*sin (1)                   4*cos(1)              4*log(1 + cos(1))       3*cos(1)*log(1 + cos(1))   4*cos (1)*log(1 + cos(1))   8*cos(1)*log(1 + cos(1))                  
- - ---------- + ------------------------ - ----------------- + ------------------------ + ------------------------ + ------------------------ - ------------------------ + ------------------------- + ------------------------ + log(1 + cos(1))
2   1 + cos(1)            2                     1 + cos(1)               2                          2                          2                        1 + cos(1)                    2                          2                                
                 2 + 2*cos (1) + 4*cos(1)                       2 + 2*cos (1) + 4*cos(1)   2 + 2*cos (1) + 4*cos(1)   2 + 2*cos (1) + 4*cos(1)                               2 + 2*cos (1) + 4*cos(1)   2 + 2*cos (1) + 4*cos(1)
$$- \frac{3}{\cos{\left(1 \right)} + 1} - \frac{3 \log{\left(\cos{\left(1 \right)} + 1 \right)}}{\cos{\left(1 \right)} + 1} - \frac{3 \log{\left(\cos{\left(1 \right)} + 1 \right)} \cos{\left(1 \right)}}{\cos{\left(1 \right)} + 1} + \frac{4 \log{\left(\cos{\left(1 \right)} + 1 \right)} \cos^{2}{\left(1 \right)}}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} + \frac{2 \sin^{2}{\left(1 \right)}}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} + \frac{4 \log{\left(\cos{\left(1 \right)} + 1 \right)}}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} + \frac{8 \log{\left(\cos{\left(1 \right)} + 1 \right)} \cos{\left(1 \right)}}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} + \log{\left(\cos{\left(1 \right)} + 1 \right)} + \frac{4 \cos{\left(1 \right)}}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} + \frac{1}{2} + \frac{4}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}}$$ 
1/2 - 3/(1 + cos(1)) + 4/(2 + 2*cos(1)^2 + 4*cos(1)) - 3*log(1 + cos(1))/(1 + cos(1)) + 2*sin(1)^2/(2 + 2*cos(1)^2 + 4*cos(1)) + 4*cos(1)/(2 + 2*cos(1)^2 + 4*cos(1)) + 4*log(1 + cos(1))/(2 + 2*cos(1)^2 + 4*cos(1)) - 3*cos(1)*log(1 + cos(1))/(1 + cos(1)) + 4*cos(1)^2*log(1 + cos(1))/(2 + 2*cos(1)^2 + 4*cos(1)) + 8*cos(1)*log(1 + cos(1))/(2 + 2*cos(1)^2 + 4*cos(1)) + log(1 + cos(1))
Respuesta numérica [src] 
0.149223205204762
0.149223205204762

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

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