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Resolución de integrales/ (3x+1)/ (3x+1)/(x^3+4)

Integral de (3x+1)/(x^3+4) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 oo           
  /           
 |            
 |  3*x + 1   
 |  ------- dx
 |    3       
 |   x  + 4   
 |            
/             
2
∫2∞3x+1x3+4 dx\int\limits_{2}^{\infty} \frac{3 x + 1}{x^{3} + 4}\, dx 
Integral((3*x + 1)/(x^3 + 4), (x, 2, oo))
Solución detallada

  1. Vuelva a escribir el integrando:

    3x+1x3+4=3xx3+4+1x3+4\frac{3 x + 1}{x^{3} + 4} = \frac{3 x}{x^{3} + 4} + \frac{1}{x^{3} + 4}

  2. Integramos término a término:

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

      ∫3xx3+4 dx=3∫xx3+4 dx\int \frac{3 x}{x^{3} + 4}\, dx = 3 \int \frac{x}{x^{3} + 4}\, dx

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

        Pero la integral

        −23log⁡(x+223)6+23log⁡(x2−223x+223)12+233atan⁡(233x3−33)6- \frac{\sqrt[3]{2} \log{\left(x + 2^{\frac{2}{3}} \right)}}{6} + \frac{\sqrt[3]{2} \log{\left(x^{2} - 2^{\frac{2}{3}} x + 2 \sqrt[3]{2} \right)}}{12} + \frac{\sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt[3]{2} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{6}

      Por lo tanto, el resultado es: −23log⁡(x+223)2+23log⁡(x2−223x+223)4+233atan⁡(233x3−33)2- \frac{\sqrt[3]{2} \log{\left(x + 2^{\frac{2}{3}} \right)}}{2} + \frac{\sqrt[3]{2} \log{\left(x^{2} - 2^{\frac{2}{3}} x + 2 \sqrt[3]{2} \right)}}{4} + \frac{\sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt[3]{2} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{2}

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

      Pero la integral

      223log⁡(x+223)12−223log⁡(x2−223x+223)24+2233atan⁡(233x3−33)12\frac{2^{\frac{2}{3}} \log{\left(x + 2^{\frac{2}{3}} \right)}}{12} - \frac{2^{\frac{2}{3}} \log{\left(x^{2} - 2^{\frac{2}{3}} x + 2 \sqrt[3]{2} \right)}}{24} + \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt[3]{2} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{12}

    El resultado es: −23log⁡(x+223)2+223log⁡(x+223)12−223log⁡(x2−223x+223)24+23log⁡(x2−223x+223)4+2233atan⁡(233x3−33)12+233atan⁡(233x3−33)2- \frac{\sqrt[3]{2} \log{\left(x + 2^{\frac{2}{3}} \right)}}{2} + \frac{2^{\frac{2}{3}} \log{\left(x + 2^{\frac{2}{3}} \right)}}{12} - \frac{2^{\frac{2}{3}} \log{\left(x^{2} - 2^{\frac{2}{3}} x + 2 \sqrt[3]{2} \right)}}{24} + \frac{\sqrt[3]{2} \log{\left(x^{2} - 2^{\frac{2}{3}} x + 2 \sqrt[3]{2} \right)}}{4} + \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt[3]{2} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{12} + \frac{\sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt[3]{2} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{2}

  3. Ahora simplificar:

    23(−12log⁡(x+223)+223log⁡(x+223)−23log⁡(x2−223x+223)+6log⁡(x2−223x+223)+2233atan⁡(3(23x−1)3)+123atan⁡(3(23x−1)3))24\frac{\sqrt[3]{2} \left(- 12 \log{\left(x + 2^{\frac{2}{3}} \right)} + 2 \sqrt[3]{2} \log{\left(x + 2^{\frac{2}{3}} \right)} - \sqrt[3]{2} \log{\left(x^{2} - 2^{\frac{2}{3}} x + 2 \sqrt[3]{2} \right)} + 6 \log{\left(x^{2} - 2^{\frac{2}{3}} x + 2 \sqrt[3]{2} \right)} + 2 \sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(\sqrt[3]{2} x - 1\right)}{3} \right)} + 12 \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(\sqrt[3]{2} x - 1\right)}{3} \right)}\right)}{24}

  4. Añadimos la constante de integración:

    23(−12log⁡(x+223)+223log⁡(x+223)−23log⁡(x2−223x+223)+6log⁡(x2−223x+223)+2233atan⁡(3(23x−1)3)+123atan⁡(3(23x−1)3))24+constant\frac{\sqrt[3]{2} \left(- 12 \log{\left(x + 2^{\frac{2}{3}} \right)} + 2 \sqrt[3]{2} \log{\left(x + 2^{\frac{2}{3}} \right)} - \sqrt[3]{2} \log{\left(x^{2} - 2^{\frac{2}{3}} x + 2 \sqrt[3]{2} \right)} + 6 \log{\left(x^{2} - 2^{\frac{2}{3}} x + 2 \sqrt[3]{2} \right)} + 2 \sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(\sqrt[3]{2} x - 1\right)}{3} \right)} + 12 \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(\sqrt[3]{2} x - 1\right)}{3} \right)}\right)}{24}+ \mathrm{constant}

Respuesta:

23(−12log⁡(x+223)+223log⁡(x+223)−23log⁡(x2−223x+223)+6log⁡(x2−223x+223)+2233atan⁡(3(23x−1)3)+123atan⁡(3(23x−1)3))24+constant\frac{\sqrt[3]{2} \left(- 12 \log{\left(x + 2^{\frac{2}{3}} \right)} + 2 \sqrt[3]{2} \log{\left(x + 2^{\frac{2}{3}} \right)} - \sqrt[3]{2} \log{\left(x^{2} - 2^{\frac{2}{3}} x + 2 \sqrt[3]{2} \right)} + 6 \log{\left(x^{2} - 2^{\frac{2}{3}} x + 2 \sqrt[3]{2} \right)} + 2 \sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(\sqrt[3]{2} x - 1\right)}{3} \right)} + 12 \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(\sqrt[3]{2} x - 1\right)}{3} \right)}\right)}{24}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                                                                                                                                                    /    ___     3 ___   ___\                  /    ___     3 ___   ___\
  /                                                                                                                                 3 ___   ___     |  \/ 3    x*\/ 2 *\/ 3 |    2/3   ___     |  \/ 3    x*\/ 2 *\/ 3 |
 |                  3 ___    /     2/3\    2/3    / 2     3 ___      2/3\   3 ___    / 2     3 ___      2/3\    2/3    /     2/3\   \/ 2 *\/ 3 *atan|- ----- + -------------|   2   *\/ 3 *atan|- ----- + -------------|
 | 3*x + 1          \/ 2 *log\x + 2   /   2   *log\x  + 2*\/ 2  - x*2   /   \/ 2 *log\x  + 2*\/ 2  - x*2   /   2   *log\x + 2   /                   \    3           3      /                  \    3           3      /
 | ------- dx = C - ------------------- - ------------------------------- + -------------------------------- + ------------------ + ----------------------------------------- + ----------------------------------------
 |   3                       2                           24                                4                           12                               2                                          12                   
 |  x  + 4                                                                                                                                                                                                              
 |                                                                                                                                                                                                                      
/
∫3x+1x3+4 dx=C−23log⁡(x+223)2+223log⁡(x+223)12−223log⁡(x2−223x+223)24+23log⁡(x2−223x+223)4+2233atan⁡(233x3−33)12+233atan⁡(233x3−33)2\int \frac{3 x + 1}{x^{3} + 4}\, dx = C - \frac{\sqrt[3]{2} \log{\left(x + 2^{\frac{2}{3}} \right)}}{2} + \frac{2^{\frac{2}{3}} \log{\left(x + 2^{\frac{2}{3}} \right)}}{12} - \frac{2^{\frac{2}{3}} \log{\left(x^{2} - 2^{\frac{2}{3}} x + 2 \sqrt[3]{2} \right)}}{24} + \frac{\sqrt[3]{2} \log{\left(x^{2} - 2^{\frac{2}{3}} x + 2 \sqrt[3]{2} \right)}}{4} + \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt[3]{2} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{12} + \frac{\sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt[3]{2} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{2}
Simplificar
Gráfica
2.00002.01002.00102.00202.00302.00402.00502.00602.00702.00802.00902-2
Trazar el gráfico f(x)
Respuesta [src] 
     /                              /                                  -pi*I     /           pi*I\         pi*I    /           5*pi*I\\            \   /                                             /          pi*I\                  /          5*pi*I\\           
     |                              |                                  ------    |           ----|         ----    |           ------||            |   |                                   -pi*I     |          ----|          pi*I    |          ------||           
     |                              | 2/3    /    3 ___  pi*I\    2/3    3       |    3 ___   3  |    2/3   3      |    3 ___    3   ||            |   |         /     2/3  pi*I\          ------    |     2/3   3  |          ----    |     2/3    3   ||           
     |                        3 ___ |2   *log\1 - \/ 2 *e    /   2   *e      *log\1 - \/ 2 *e    /   2   *e    *log\1 - \/ 2 *e      /|            |   |3 ___    |    2   *e    |   3 ___    3       |    2   *e    |   3 ___   3      |    2   *e      ||           
     |                        \/ 2 *|------------------------- - --------------------------------- - ---------------------------------|*Gamma(-1/3)|   |\/ 2 *log|1 - ----------|   \/ 2 *e      *log|1 - ----------|   \/ 2 *e    *log|1 - ------------||           
 2/3 |                              \            6                               6                                   6                /            |   |         \        2     /                    \        2     /                  \         2      /|           
2   *|Gamma(1/3)*Gamma(2/3) + ---------------------------------------------------------------------------------------------------------------------|   |------------------------- - --------------------------------- - ---------------------------------|*Gamma(1/3)
     \                                                                              Gamma(2/3)                                                     /   \            3                               3                                   3                /           
---------------------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------
                                                                         12                                                                                                                             2*Gamma(4/3)
223(23(−223eiπ3log⁡(−23e5iπ3+1)6+223log⁡(1−23eiπ)6−223e−iπ3log⁡(1−23eiπ3)6)Γ(−13)Γ(23)+Γ(13)Γ(23))12+(−23eiπ3log⁡(−223e5iπ32+1)3+23log⁡(−223eiπ2+1)3−23e−iπ3log⁡(1−223eiπ32)3)Γ(13)2Γ(43)\frac{2^{\frac{2}{3}} \left(\frac{\sqrt[3]{2} \left(- \frac{2^{\frac{2}{3}} e^{\frac{i \pi}{3}} \log{\left(- \sqrt[3]{2} e^{\frac{5 i \pi}{3}} + 1 \right)}}{6} + \frac{2^{\frac{2}{3}} \log{\left(1 - \sqrt[3]{2} e^{i \pi} \right)}}{6} - \frac{2^{\frac{2}{3}} e^{- \frac{i \pi}{3}} \log{\left(1 - \sqrt[3]{2} e^{\frac{i \pi}{3}} \right)}}{6}\right) \Gamma\left(- \frac{1}{3}\right)}{\Gamma\left(\frac{2}{3}\right)} + \Gamma\left(\frac{1}{3}\right) \Gamma\left(\frac{2}{3}\right)\right)}{12} + \frac{\left(- \frac{\sqrt[3]{2} e^{\frac{i \pi}{3}} \log{\left(- \frac{2^{\frac{2}{3}} e^{\frac{5 i \pi}{3}}}{2} + 1 \right)}}{3} + \frac{\sqrt[3]{2} \log{\left(- \frac{2^{\frac{2}{3}} e^{i \pi}}{2} + 1 \right)}}{3} - \frac{\sqrt[3]{2} e^{- \frac{i \pi}{3}} \log{\left(1 - \frac{2^{\frac{2}{3}} e^{\frac{i \pi}{3}}}{2} \right)}}{3}\right) \Gamma\left(\frac{1}{3}\right)}{2 \Gamma\left(\frac{4}{3}\right)} 
=
= 
     /                              /                                  -pi*I     /           pi*I\         pi*I    /           5*pi*I\\            \   /                                             /          pi*I\                  /          5*pi*I\\           
     |                              |                                  ------    |           ----|         ----    |           ------||            |   |                                   -pi*I     |          ----|          pi*I    |          ------||           
     |                              | 2/3    /    3 ___  pi*I\    2/3    3       |    3 ___   3  |    2/3   3      |    3 ___    3   ||            |   |         /     2/3  pi*I\          ------    |     2/3   3  |          ----    |     2/3    3   ||           
     |                        3 ___ |2   *log\1 - \/ 2 *e    /   2   *e      *log\1 - \/ 2 *e    /   2   *e    *log\1 - \/ 2 *e      /|            |   |3 ___    |    2   *e    |   3 ___    3       |    2   *e    |   3 ___   3      |    2   *e      ||           
     |                        \/ 2 *|------------------------- - --------------------------------- - ---------------------------------|*Gamma(-1/3)|   |\/ 2 *log|1 - ----------|   \/ 2 *e      *log|1 - ----------|   \/ 2 *e    *log|1 - ------------||           
 2/3 |                              \            6                               6                                   6                /            |   |         \        2     /                    \        2     /                  \         2      /|           
2   *|Gamma(1/3)*Gamma(2/3) + ---------------------------------------------------------------------------------------------------------------------|   |------------------------- - --------------------------------- - ---------------------------------|*Gamma(1/3)
     \                                                                              Gamma(2/3)                                                     /   \            3                               3                                   3                /           
---------------------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------
                                                                         12                                                                                                                             2*Gamma(4/3)
223(23(−223eiπ3log⁡(−23e5iπ3+1)6+223log⁡(1−23eiπ)6−223e−iπ3log⁡(1−23eiπ3)6)Γ(−13)Γ(23)+Γ(13)Γ(23))12+(−23eiπ3log⁡(−223e5iπ32+1)3+23log⁡(−223eiπ2+1)3−23e−iπ3log⁡(1−223eiπ32)3)Γ(13)2Γ(43)\frac{2^{\frac{2}{3}} \left(\frac{\sqrt[3]{2} \left(- \frac{2^{\frac{2}{3}} e^{\frac{i \pi}{3}} \log{\left(- \sqrt[3]{2} e^{\frac{5 i \pi}{3}} + 1 \right)}}{6} + \frac{2^{\frac{2}{3}} \log{\left(1 - \sqrt[3]{2} e^{i \pi} \right)}}{6} - \frac{2^{\frac{2}{3}} e^{- \frac{i \pi}{3}} \log{\left(1 - \sqrt[3]{2} e^{\frac{i \pi}{3}} \right)}}{6}\right) \Gamma\left(- \frac{1}{3}\right)}{\Gamma\left(\frac{2}{3}\right)} + \Gamma\left(\frac{1}{3}\right) \Gamma\left(\frac{2}{3}\right)\right)}{12} + \frac{\left(- \frac{\sqrt[3]{2} e^{\frac{i \pi}{3}} \log{\left(- \frac{2^{\frac{2}{3}} e^{\frac{5 i \pi}{3}}}{2} + 1 \right)}}{3} + \frac{\sqrt[3]{2} \log{\left(- \frac{2^{\frac{2}{3}} e^{i \pi}}{2} + 1 \right)}}{3} - \frac{\sqrt[3]{2} e^{- \frac{i \pi}{3}} \log{\left(1 - \frac{2^{\frac{2}{3}} e^{\frac{i \pi}{3}}}{2} \right)}}{3}\right) \Gamma\left(\frac{1}{3}\right)}{2 \Gamma\left(\frac{4}{3}\right)} 
2^(2/3)*(gamma(1/3)*gamma(2/3) + 2^(1/3)*(2^(2/3)*log(1 - 2^(1/3)*exp_polar(pi*i))/6 - 2^(2/3)*exp(-pi*i/3)*log(1 - 2^(1/3)*exp_polar(pi*i/3))/6 - 2^(2/3)*exp(pi*i/3)*log(1 - 2^(1/3)*exp_polar(5*pi*i/3))/6)*gamma(-1/3)/gamma(2/3))/12 + (2^(1/3)*log(1 - 2^(2/3)*exp_polar(pi*i)/2)/3 - 2^(1/3)*exp(-pi*i/3)*log(1 - 2^(2/3)*exp_polar(pi*i/3)/2)/3 - 2^(1/3)*exp(pi*i/3)*log(1 - 2^(2/3)*exp_polar(5*pi*i/3)/2)/3)*gamma(1/3)/(2*gamma(4/3))

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

    © Sr Examen – Калькулятор