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Integral de -(4*x^(3/2)+1)/(x^3+4) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
 oo                
  /                
 |                 
 |       3/2       
 |  - 4*x    - 1   
 |  ------------ dx
 |      3          
 |     x  + 4      
 |                 
/                  
3                  
34x321x3+4dx\int\limits_{3}^{\infty} \frac{- 4 x^{\frac{3}{2}} - 1}{x^{3} + 4}\, dx
Integral((-4*x^(3/2) - 1)/(x^3 + 4), (x, 3, oo))
Solución detallada
  1. Hay varias maneras de calcular esta integral.

    Método #1

    1. Vuelva a escribir el integrando:

      4x321x3+4=4x32+1x3+4\frac{- 4 x^{\frac{3}{2}} - 1}{x^{3} + 4} = - \frac{4 x^{\frac{3}{2}} + 1}{x^{3} + 4}

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

      (4x32+1x3+4)dx=4x32+1x3+4dx\int \left(- \frac{4 x^{\frac{3}{2}} + 1}{x^{3} + 4}\right)\, dx = - \int \frac{4 x^{\frac{3}{2}} + 1}{x^{3} + 4}\, dx

      1. Vuelva a escribir el integrando:

        4x32+1x3+4=4x32x3+4+1x3+4\frac{4 x^{\frac{3}{2}} + 1}{x^{3} + 4} = \frac{4 x^{\frac{3}{2}}}{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:

          4x32x3+4dx=4x32x3+4dx\int \frac{4 x^{\frac{3}{2}}}{x^{3} + 4}\, dx = 4 \int \frac{x^{\frac{3}{2}}}{x^{3} + 4}\, dx

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

            Pero la integral

            2233log(4233x+4x+4223)122233log(4233x+4x+4223)12+223atan(223x2)3+223atan(223x3)6+223atan(223x+3)6\frac{2^{\frac{2}{3}} \sqrt{3} \log{\left(- 4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{12} - \frac{2^{\frac{2}{3}} \sqrt{3} \log{\left(4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{12} + \frac{2^{\frac{2}{3}} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{x}}{2} \right)}}{3} + \frac{2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} - \sqrt{3} \right)}}{6} + \frac{2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} + \sqrt{3} \right)}}{6}

          Por lo tanto, el resultado es: 2233log(4233x+4x+4223)32233log(4233x+4x+4223)3+4223atan(223x2)3+2223atan(223x3)3+2223atan(223x+3)3\frac{2^{\frac{2}{3}} \sqrt{3} \log{\left(- 4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{3} - \frac{2^{\frac{2}{3}} \sqrt{3} \log{\left(4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{3} + \frac{4 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{x}}{2} \right)}}{3} + \frac{2 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} - \sqrt{3} \right)}}{3} + \frac{2 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} + \sqrt{3} \right)}}{3}

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

          Pero la integral

          223log(x+223)12223log(x2223x+223)24+2233atan(233x333)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: 223log(x+223)12223log(x2223x+223)24+2233log(4233x+4x+4223)32233log(4233x+4x+4223)3+4223atan(223x2)3+2223atan(223x3)3+2223atan(223x+3)3+2233atan(233x333)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} \log{\left(- 4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{3} - \frac{2^{\frac{2}{3}} \sqrt{3} \log{\left(4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{3} + \frac{4 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{x}}{2} \right)}}{3} + \frac{2 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} - \sqrt{3} \right)}}{3} + \frac{2 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} + \sqrt{3} \right)}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt[3]{2} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{12}

      Por lo tanto, el resultado es: 223log(x+223)12+223log(x2223x+223)242233log(4233x+4x+4223)3+2233log(4233x+4x+4223)34223atan(223x2)32223atan(223x3)32223atan(223x+3)32233atan(233x333)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} \log{\left(- 4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3} \log{\left(4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{3} - \frac{4 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{x}}{2} \right)}}{3} - \frac{2 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} - \sqrt{3} \right)}}{3} - \frac{2 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} + \sqrt{3} \right)}}{3} - \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt[3]{2} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{12}

    Método #2

    1. Vuelva a escribir el integrando:

      4x321x3+4=4x32x3+41x3+4\frac{- 4 x^{\frac{3}{2}} - 1}{x^{3} + 4} = - \frac{4 x^{\frac{3}{2}}}{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:

        (4x32x3+4)dx=4x32x3+4dx\int \left(- \frac{4 x^{\frac{3}{2}}}{x^{3} + 4}\right)\, dx = - 4 \int \frac{x^{\frac{3}{2}}}{x^{3} + 4}\, dx

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

          Pero la integral

          2233log(4233x+4x+4223)122233log(4233x+4x+4223)12+223atan(223x2)3+223atan(223x3)6+223atan(223x+3)6\frac{2^{\frac{2}{3}} \sqrt{3} \log{\left(- 4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{12} - \frac{2^{\frac{2}{3}} \sqrt{3} \log{\left(4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{12} + \frac{2^{\frac{2}{3}} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{x}}{2} \right)}}{3} + \frac{2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} - \sqrt{3} \right)}}{6} + \frac{2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} + \sqrt{3} \right)}}{6}

        Por lo tanto, el resultado es: 2233log(4233x+4x+4223)3+2233log(4233x+4x+4223)34223atan(223x2)32223atan(223x3)32223atan(223x+3)3- \frac{2^{\frac{2}{3}} \sqrt{3} \log{\left(- 4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3} \log{\left(4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{3} - \frac{4 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{x}}{2} \right)}}{3} - \frac{2 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} - \sqrt{3} \right)}}{3} - \frac{2 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} + \sqrt{3} \right)}}{3}

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

        (1x3+4)dx=1x3+4dx\int \left(- \frac{1}{x^{3} + 4}\right)\, dx = - \int \frac{1}{x^{3} + 4}\, dx

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

          Pero la integral

          223log(x+223)12223log(x2223x+223)24+2233atan(233x333)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}

        Por lo tanto, el resultado es: 223log(x+223)12+223log(x2223x+223)242233atan(233x333)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: 223log(x+223)12+223log(x2223x+223)242233log(4233x+4x+4223)3+2233log(4233x+4x+4223)34223atan(223x2)32223atan(223x3)32223atan(223x+3)32233atan(233x333)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} \log{\left(- 4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3} \log{\left(4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{3} - \frac{4 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{x}}{2} \right)}}{3} - \frac{2 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} - \sqrt{3} \right)}}{3} - \frac{2 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} + \sqrt{3} \right)}}{3} - \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt[3]{2} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{12}

  2. Ahora simplificar:

    223(2log(x+223)+log(x2223x+223)83log(233x+x+223)+83log(233x+x+223)32atan(223x2)23atan(3(23x1)3)16atan(223x3)16atan(223x+3))24\frac{2^{\frac{2}{3}} \left(- 2 \log{\left(x + 2^{\frac{2}{3}} \right)} + \log{\left(x^{2} - 2^{\frac{2}{3}} x + 2 \sqrt[3]{2} \right)} - 8 \sqrt{3} \log{\left(- \sqrt[3]{2} \sqrt{3} \sqrt{x} + x + 2^{\frac{2}{3}} \right)} + 8 \sqrt{3} \log{\left(\sqrt[3]{2} \sqrt{3} \sqrt{x} + x + 2^{\frac{2}{3}} \right)} - 32 \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{x}}{2} \right)} - 2 \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(\sqrt[3]{2} x - 1\right)}{3} \right)} - 16 \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} - \sqrt{3} \right)} - 16 \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} + \sqrt{3} \right)}\right)}{24}

  3. Añadimos la constante de integración:

    223(2log(x+223)+log(x2223x+223)83log(233x+x+223)+83log(233x+x+223)32atan(223x2)23atan(3(23x1)3)16atan(223x3)16atan(223x+3))24+constant\frac{2^{\frac{2}{3}} \left(- 2 \log{\left(x + 2^{\frac{2}{3}} \right)} + \log{\left(x^{2} - 2^{\frac{2}{3}} x + 2 \sqrt[3]{2} \right)} - 8 \sqrt{3} \log{\left(- \sqrt[3]{2} \sqrt{3} \sqrt{x} + x + 2^{\frac{2}{3}} \right)} + 8 \sqrt{3} \log{\left(\sqrt[3]{2} \sqrt{3} \sqrt{x} + x + 2^{\frac{2}{3}} \right)} - 32 \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{x}}{2} \right)} - 2 \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(\sqrt[3]{2} x - 1\right)}{3} \right)} - 16 \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} - \sqrt{3} \right)} - 16 \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} + \sqrt{3} \right)}\right)}{24}+ \mathrm{constant}


Respuesta:

223(2log(x+223)+log(x2223x+223)83log(233x+x+223)+83log(233x+x+223)32atan(223x2)23atan(3(23x1)3)16atan(223x3)16atan(223x+3))24+constant\frac{2^{\frac{2}{3}} \left(- 2 \log{\left(x + 2^{\frac{2}{3}} \right)} + \log{\left(x^{2} - 2^{\frac{2}{3}} x + 2 \sqrt[3]{2} \right)} - 8 \sqrt{3} \log{\left(- \sqrt[3]{2} \sqrt{3} \sqrt{x} + x + 2^{\frac{2}{3}} \right)} + 8 \sqrt{3} \log{\left(\sqrt[3]{2} \sqrt{3} \sqrt{x} + x + 2^{\frac{2}{3}} \right)} - 32 \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{x}}{2} \right)} - 2 \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(\sqrt[3]{2} x - 1\right)}{3} \right)} - 16 \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} - \sqrt{3} \right)} - 16 \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} + \sqrt{3} \right)}\right)}{24}+ \mathrm{constant}

Respuesta (Indefinida) [src]
  /                                 / 2/3   ___\                                                                                                                                                                                                    /    ___     3 ___   ___\                                                     
 |                          2/3     |2   *\/ x |                                                                                                                                                                                      2/3   ___     |  \/ 3    x*\/ 2 *\/ 3 |                                                     
 |      3/2              4*2   *atan|----------|      2/3     /  ___    2/3   ___\      2/3     /    ___    2/3   ___\    2/3    /     2/3\    2/3    / 2     3 ___      2/3\    2/3   ___    /         2/3     3 ___   ___   ___\   2   *\/ 3 *atan|- ----- + -------------|    2/3   ___    /         2/3     3 ___   ___   ___\
 | - 4*x    - 1                     \    2     /   2*2   *atan\\/ 3  + 2   *\/ x /   2*2   *atan\- \/ 3  + 2   *\/ x /   2   *log\x + 2   /   2   *log\x  + 2*\/ 2  - x*2   /   2   *\/ 3 *log\4*x + 4*2    - 4*\/ 2 *\/ 3 *\/ x /                  \    3           3      /   2   *\/ 3 *log\4*x + 4*2    + 4*\/ 2 *\/ 3 *\/ x /
 | ------------ dx = C - ----------------------- - ------------------------------- - --------------------------------- - ------------------ + ------------------------------- - -------------------------------------------------- - ---------------------------------------- + --------------------------------------------------
 |     3                            3                             3                                  3                           12                          24                                         3                                               12                                              3                         
 |    x  + 4                                                                                                                                                                                                                                                                                                                      
 |                                                                                                                                                                                                                                                                                                                                
/                                                                                                                                                                                                                                                                                                                                 
4x321x3+4dx=C223log(x+223)12+223log(x2223x+223)242233log(4233x+4x+4223)3+2233log(4233x+4x+4223)34223atan(223x2)32223atan(223x3)32223atan(223x+3)32233atan(233x333)12\int \frac{- 4 x^{\frac{3}{2}} - 1}{x^{3} + 4}\, dx = C - \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} \log{\left(- 4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3} \log{\left(4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{3} - \frac{4 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{x}}{2} \right)}}{3} - \frac{2 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} - \sqrt{3} \right)}}{3} - \frac{2 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} + \sqrt{3} \right)}}{3} - \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt[3]{2} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{12}

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