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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1               
  /               
 |                
 |     3/    x\   
 |  sin |1 - -|   
 |      \    3/   
 |  ----------- dx
 |      /    x\   
 |   cos|1 - -|   
 |      \    3/   
 |                
/                 
0
$$\int\limits_{0}^{1} \frac{\sin^{3}{\left(- \frac{x}{3} + 1 \right)}}{\cos{\left(- \frac{x}{3} + 1 \right)}}\, dx$$ 
Integral(sin(1 - x/3)^3/cos(1 - x/3), (x, 0, 1))
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
               2                                                                                      /       2     \               /       2     \                                                                                2                     2                                        2                               2         /       2     \        4                                        4                               4         /       2     \        4         /       2     \        4                                        4                               2         /       2     \        2                                        2                       
          6*tan (1/2)           3*(pi*I + log(1 + tan(1/2)))       3*log(1 - tan(1/2))           3*log\1 + tan (1/3)/          3*log\1 + tan (1/2)/      3*(pi*I + log(1 + tan(1/3)))       3*log(1 - tan(1/3))               6*tan (1/3)           6*tan (1/2)*(pi*I + log(1 + tan(1/2)))   6*tan (1/2)*log(1 - tan(1/2))   6*tan (1/3)*log\1 + tan (1/3)/   3*tan (1/2)*(pi*I + log(1 + tan(1/2)))   3*tan (1/2)*log(1 - tan(1/2))   3*tan (1/3)*log\1 + tan (1/3)/   3*tan (1/2)*log\1 + tan (1/2)/   3*tan (1/3)*(pi*I + log(1 + tan(1/3)))   3*tan (1/3)*log(1 - tan(1/3))   6*tan (1/2)*log\1 + tan (1/2)/   6*tan (1/3)*(pi*I + log(1 + tan(1/3)))   6*tan (1/3)*log(1 - tan(1/3))
- --------------------------- - ---------------------------- - --------------------------- - --------------------------- + --------------------------- + ---------------------------- + --------------------------- + --------------------------- - -------------------------------------- - ----------------------------- - ------------------------------ - -------------------------------------- - ----------------------------- - ------------------------------ + ------------------------------ + -------------------------------------- + ----------------------------- + ------------------------------ + -------------------------------------- + -----------------------------
         4             2               4             2                4             2               4             2               4             2               4             2                4             2               4             2                    4             2                      4             2                 4             2                      4             2                      4             2                 4             2                  4             2                      4             2                      4             2                 4             2                      4             2                      4             2      
  1 + tan (1/2) + 2*tan (1/2)   1 + tan (1/2) + 2*tan (1/2)    1 + tan (1/2) + 2*tan (1/2)   1 + tan (1/3) + 2*tan (1/3)   1 + tan (1/2) + 2*tan (1/2)   1 + tan (1/3) + 2*tan (1/3)    1 + tan (1/3) + 2*tan (1/3)   1 + tan (1/3) + 2*tan (1/3)        1 + tan (1/2) + 2*tan (1/2)          1 + tan (1/2) + 2*tan (1/2)     1 + tan (1/3) + 2*tan (1/3)          1 + tan (1/2) + 2*tan (1/2)          1 + tan (1/2) + 2*tan (1/2)     1 + tan (1/3) + 2*tan (1/3)      1 + tan (1/2) + 2*tan (1/2)          1 + tan (1/3) + 2*tan (1/3)          1 + tan (1/3) + 2*tan (1/3)     1 + tan (1/2) + 2*tan (1/2)          1 + tan (1/3) + 2*tan (1/3)          1 + tan (1/3) + 2*tan (1/3)
$$- \frac{6 \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{3 \log{\left(1 - \tan{\left(\frac{1}{3} \right)} \right)}}{\tan^{4}{\left(\frac{1}{3} \right)} + 2 \tan^{2}{\left(\frac{1}{3} \right)} + 1} - \frac{3 \log{\left(\tan^{2}{\left(\frac{1}{3} \right)} + 1 \right)}}{\tan^{4}{\left(\frac{1}{3} \right)} + 2 \tan^{2}{\left(\frac{1}{3} \right)} + 1} + \frac{6 \log{\left(1 - \tan{\left(\frac{1}{3} \right)} \right)} \tan^{2}{\left(\frac{1}{3} \right)}}{\tan^{4}{\left(\frac{1}{3} \right)} + 2 \tan^{2}{\left(\frac{1}{3} \right)} + 1} - \frac{6 \log{\left(\tan^{2}{\left(\frac{1}{3} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{3} \right)}}{\tan^{4}{\left(\frac{1}{3} \right)} + 2 \tan^{2}{\left(\frac{1}{3} \right)} + 1} + \frac{3 \log{\left(1 - \tan{\left(\frac{1}{3} \right)} \right)} \tan^{4}{\left(\frac{1}{3} \right)}}{\tan^{4}{\left(\frac{1}{3} \right)} + 2 \tan^{2}{\left(\frac{1}{3} \right)} + 1} - \frac{3 \log{\left(\tan^{2}{\left(\frac{1}{3} \right)} + 1 \right)} \tan^{4}{\left(\frac{1}{3} \right)}}{\tan^{4}{\left(\frac{1}{3} \right)} + 2 \tan^{2}{\left(\frac{1}{3} \right)} + 1} + \frac{3 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{3 \log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} \tan^{4}{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{6 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{3 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{6 \tan^{2}{\left(\frac{1}{3} \right)}}{\tan^{4}{\left(\frac{1}{3} \right)} + 2 \tan^{2}{\left(\frac{1}{3} \right)} + 1} - \frac{6 \log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{3 \log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{3 \left(\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right)}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{6 \left(\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{3 \left(\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right) \tan^{4}{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{3 \left(\log{\left(\tan{\left(\frac{1}{3} \right)} + 1 \right)} + i \pi\right) \tan^{4}{\left(\frac{1}{3} \right)}}{\tan^{4}{\left(\frac{1}{3} \right)} + 2 \tan^{2}{\left(\frac{1}{3} \right)} + 1} + \frac{6 \left(\log{\left(\tan{\left(\frac{1}{3} \right)} + 1 \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{3} \right)}}{\tan^{4}{\left(\frac{1}{3} \right)} + 2 \tan^{2}{\left(\frac{1}{3} \right)} + 1} + \frac{3 \left(\log{\left(\tan{\left(\frac{1}{3} \right)} + 1 \right)} + i \pi\right)}{\tan^{4}{\left(\frac{1}{3} \right)} + 2 \tan^{2}{\left(\frac{1}{3} \right)} + 1}$$ 
=
= 
               2                                                                                      /       2     \               /       2     \                                                                                2                     2                                        2                               2         /       2     \        4                                        4                               4         /       2     \        4         /       2     \        4                                        4                               2         /       2     \        2                                        2                       
          6*tan (1/2)           3*(pi*I + log(1 + tan(1/2)))       3*log(1 - tan(1/2))           3*log\1 + tan (1/3)/          3*log\1 + tan (1/2)/      3*(pi*I + log(1 + tan(1/3)))       3*log(1 - tan(1/3))               6*tan (1/3)           6*tan (1/2)*(pi*I + log(1 + tan(1/2)))   6*tan (1/2)*log(1 - tan(1/2))   6*tan (1/3)*log\1 + tan (1/3)/   3*tan (1/2)*(pi*I + log(1 + tan(1/2)))   3*tan (1/2)*log(1 - tan(1/2))   3*tan (1/3)*log\1 + tan (1/3)/   3*tan (1/2)*log\1 + tan (1/2)/   3*tan (1/3)*(pi*I + log(1 + tan(1/3)))   3*tan (1/3)*log(1 - tan(1/3))   6*tan (1/2)*log\1 + tan (1/2)/   6*tan (1/3)*(pi*I + log(1 + tan(1/3)))   6*tan (1/3)*log(1 - tan(1/3))
- --------------------------- - ---------------------------- - --------------------------- - --------------------------- + --------------------------- + ---------------------------- + --------------------------- + --------------------------- - -------------------------------------- - ----------------------------- - ------------------------------ - -------------------------------------- - ----------------------------- - ------------------------------ + ------------------------------ + -------------------------------------- + ----------------------------- + ------------------------------ + -------------------------------------- + -----------------------------
         4             2               4             2                4             2               4             2               4             2               4             2                4             2               4             2                    4             2                      4             2                 4             2                      4             2                      4             2                 4             2                  4             2                      4             2                      4             2                 4             2                      4             2                      4             2      
  1 + tan (1/2) + 2*tan (1/2)   1 + tan (1/2) + 2*tan (1/2)    1 + tan (1/2) + 2*tan (1/2)   1 + tan (1/3) + 2*tan (1/3)   1 + tan (1/2) + 2*tan (1/2)   1 + tan (1/3) + 2*tan (1/3)    1 + tan (1/3) + 2*tan (1/3)   1 + tan (1/3) + 2*tan (1/3)        1 + tan (1/2) + 2*tan (1/2)          1 + tan (1/2) + 2*tan (1/2)     1 + tan (1/3) + 2*tan (1/3)          1 + tan (1/2) + 2*tan (1/2)          1 + tan (1/2) + 2*tan (1/2)     1 + tan (1/3) + 2*tan (1/3)      1 + tan (1/2) + 2*tan (1/2)          1 + tan (1/3) + 2*tan (1/3)          1 + tan (1/3) + 2*tan (1/3)     1 + tan (1/2) + 2*tan (1/2)          1 + tan (1/3) + 2*tan (1/3)          1 + tan (1/3) + 2*tan (1/3)
$$- \frac{6 \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{3 \log{\left(1 - \tan{\left(\frac{1}{3} \right)} \right)}}{\tan^{4}{\left(\frac{1}{3} \right)} + 2 \tan^{2}{\left(\frac{1}{3} \right)} + 1} - \frac{3 \log{\left(\tan^{2}{\left(\frac{1}{3} \right)} + 1 \right)}}{\tan^{4}{\left(\frac{1}{3} \right)} + 2 \tan^{2}{\left(\frac{1}{3} \right)} + 1} + \frac{6 \log{\left(1 - \tan{\left(\frac{1}{3} \right)} \right)} \tan^{2}{\left(\frac{1}{3} \right)}}{\tan^{4}{\left(\frac{1}{3} \right)} + 2 \tan^{2}{\left(\frac{1}{3} \right)} + 1} - \frac{6 \log{\left(\tan^{2}{\left(\frac{1}{3} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{3} \right)}}{\tan^{4}{\left(\frac{1}{3} \right)} + 2 \tan^{2}{\left(\frac{1}{3} \right)} + 1} + \frac{3 \log{\left(1 - \tan{\left(\frac{1}{3} \right)} \right)} \tan^{4}{\left(\frac{1}{3} \right)}}{\tan^{4}{\left(\frac{1}{3} \right)} + 2 \tan^{2}{\left(\frac{1}{3} \right)} + 1} - \frac{3 \log{\left(\tan^{2}{\left(\frac{1}{3} \right)} + 1 \right)} \tan^{4}{\left(\frac{1}{3} \right)}}{\tan^{4}{\left(\frac{1}{3} \right)} + 2 \tan^{2}{\left(\frac{1}{3} \right)} + 1} + \frac{3 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{3 \log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} \tan^{4}{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{6 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{3 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{6 \tan^{2}{\left(\frac{1}{3} \right)}}{\tan^{4}{\left(\frac{1}{3} \right)} + 2 \tan^{2}{\left(\frac{1}{3} \right)} + 1} - \frac{6 \log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{3 \log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{3 \left(\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right)}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{6 \left(\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{3 \left(\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right) \tan^{4}{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{3 \left(\log{\left(\tan{\left(\frac{1}{3} \right)} + 1 \right)} + i \pi\right) \tan^{4}{\left(\frac{1}{3} \right)}}{\tan^{4}{\left(\frac{1}{3} \right)} + 2 \tan^{2}{\left(\frac{1}{3} \right)} + 1} + \frac{6 \left(\log{\left(\tan{\left(\frac{1}{3} \right)} + 1 \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{3} \right)}}{\tan^{4}{\left(\frac{1}{3} \right)} + 2 \tan^{2}{\left(\frac{1}{3} \right)} + 1} + \frac{3 \left(\log{\left(\tan{\left(\frac{1}{3} \right)} + 1 \right)} + i \pi\right)}{\tan^{4}{\left(\frac{1}{3} \right)} + 2 \tan^{2}{\left(\frac{1}{3} \right)} + 1}$$ 
-6*tan(1/2)^2/(1 + tan(1/2)^4 + 2*tan(1/2)^2) - 3*(pi*i + log(1 + tan(1/2)))/(1 + tan(1/2)^4 + 2*tan(1/2)^2) - 3*log(1 - tan(1/2))/(1 + tan(1/2)^4 + 2*tan(1/2)^2) - 3*log(1 + tan(1/3)^2)/(1 + tan(1/3)^4 + 2*tan(1/3)^2) + 3*log(1 + tan(1/2)^2)/(1 + tan(1/2)^4 + 2*tan(1/2)^2) + 3*(pi*i + log(1 + tan(1/3)))/(1 + tan(1/3)^4 + 2*tan(1/3)^2) + 3*log(1 - tan(1/3))/(1 + tan(1/3)^4 + 2*tan(1/3)^2) + 6*tan(1/3)^2/(1 + tan(1/3)^4 + 2*tan(1/3)^2) - 6*tan(1/2)^2*(pi*i + log(1 + tan(1/2)))/(1 + tan(1/2)^4 + 2*tan(1/2)^2) - 6*tan(1/2)^2*log(1 - tan(1/2))/(1 + tan(1/2)^4 + 2*tan(1/2)^2) - 6*tan(1/3)^2*log(1 + tan(1/3)^2)/(1 + tan(1/3)^4 + 2*tan(1/3)^2) - 3*tan(1/2)^4*(pi*i + log(1 + tan(1/2)))/(1 + tan(1/2)^4 + 2*tan(1/2)^2) - 3*tan(1/2)^4*log(1 - tan(1/2))/(1 + tan(1/2)^4 + 2*tan(1/2)^2) - 3*tan(1/3)^4*log(1 + tan(1/3)^2)/(1 + tan(1/3)^4 + 2*tan(1/3)^2) + 3*tan(1/2)^4*log(1 + tan(1/2)^2)/(1 + tan(1/2)^4 + 2*tan(1/2)^2) + 3*tan(1/3)^4*(pi*i + log(1 + tan(1/3)))/(1 + tan(1/3)^4 + 2*tan(1/3)^2) + 3*tan(1/3)^4*log(1 - tan(1/3))/(1 + tan(1/3)^4 + 2*tan(1/3)^2) + 6*tan(1/2)^2*log(1 + tan(1/2)^2)/(1 + tan(1/2)^4 + 2*tan(1/2)^2) + 6*tan(1/3)^2*(pi*i + log(1 + tan(1/3)))/(1 + tan(1/3)^4 + 2*tan(1/3)^2) + 6*tan(1/3)^2*log(1 - tan(1/3))/(1 + tan(1/3)^4 + 2*tan(1/3)^2)
Respuesta numérica [src] 
0.635515310870617
0.635515310870617

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

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