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Expresiones semejantes
x^2-1/(x^3+3x+1)^2
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Resolución de integrales/ x^2+1/ x^3+3/ x^2+1/(x^3+3x+1)^2

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

Solución

Ha introducido [src]

 oo                          
  /                          
 |                           
 |  / 2          1       \   
 |  |x  + ---------------| dx
 |  |                   2|   
 |  |     / 3          \ |   
 |  \     \x  + 3*x + 1/ /   
 |                           
/                            
0

$$\int\limits_{0}^{\infty} \left(x^{2} + \frac{1}{\left(\left(x^{3} + 3 x\right) + 1\right)^{2}}\right)\, dx$$

Integral(x^2 + 1/((x^3 + 3*x + 1)^2), (x, 0, oo))

Solución detallada

Integramos término a término:
1. Integral $x^{n}$ es $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int x^{2}\, dx = \frac{x^{3}}{3}$
1. No puedo encontrar los pasos en la búsqueda de esta integral.
  
  Pero la integral
  
  $- \frac{2 x^{2} - x + 4}{15 x^{3} + 45 x + 15} + \operatorname{RootSum} {\left(91125 t^{3} - 108 t - 8, \left( t \mapsto t \log{\left(\frac{30375 t^{2}}{44} - \frac{225 t}{11} + x - \frac{6}{11} \right)} \right)\right)}$
El resultado es: $\frac{x^{3}}{3} - \frac{2 x^{2} - x + 4}{15 x^{3} + 45 x + 15} + \operatorname{RootSum} {\left(91125 t^{3} - 108 t - 8, \left( t \mapsto t \log{\left(\frac{30375 t^{2}}{44} - \frac{225 t}{11} + x - \frac{6}{11} \right)} \right)\right)}$
Ahora simplificar:

$\frac{- 2 x^{2} + x + 5 \left(x^{3} + 3 x + 1\right) \left(x^{3} + 3 \left(\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}} + \frac{4}{10125 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}\right) \log{\left(x - \frac{6}{11} - \frac{4}{495 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} - \frac{225 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}{11} + \frac{30375 \left(\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}} + \frac{4}{10125 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}\right)^{2}}{44} \right)} + 3 \left(\frac{4}{10125 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}\right) \log{\left(x - \frac{6}{11} + \frac{30375 \left(\frac{4}{10125 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}\right)^{2}}{44} - \frac{225 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}{11} - \frac{4}{495 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} \right)} + 3 \left(\frac{4}{10125 \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}\right) \log{\left(x - \frac{225 \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}{11} - \frac{6}{11} - \frac{4}{495 \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \frac{30375 \left(\frac{4}{10125 \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}\right)^{2}}{44} \right)}\right) - 4}{15 \left(x^{3} + 3 x + 1\right)}$
Añadimos la constante de integración:

$\frac{- 2 x^{2} + x + 5 \left(x^{3} + 3 x + 1\right) \left(x^{3} + 3 \left(\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}} + \frac{4}{10125 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}\right) \log{\left(x - \frac{6}{11} - \frac{4}{495 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} - \frac{225 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}{11} + \frac{30375 \left(\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}} + \frac{4}{10125 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}\right)^{2}}{44} \right)} + 3 \left(\frac{4}{10125 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}\right) \log{\left(x - \frac{6}{11} + \frac{30375 \left(\frac{4}{10125 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}\right)^{2}}{44} - \frac{225 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}{11} - \frac{4}{495 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} \right)} + 3 \left(\frac{4}{10125 \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}\right) \log{\left(x - \frac{225 \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}{11} - \frac{6}{11} - \frac{4}{495 \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \frac{30375 \left(\frac{4}{10125 \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}\right)^{2}}{44} \right)}\right) - 4}{15 \left(x^{3} + 3 x + 1\right)}+ \mathrm{constant}$

Respuesta:

$\frac{- 2 x^{2} + x + 5 \left(x^{3} + 3 x + 1\right) \left(x^{3} + 3 \left(\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}} + \frac{4}{10125 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}\right) \log{\left(x - \frac{6}{11} - \frac{4}{495 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} - \frac{225 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}{11} + \frac{30375 \left(\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}} + \frac{4}{10125 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}\right)^{2}}{44} \right)} + 3 \left(\frac{4}{10125 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}\right) \log{\left(x - \frac{6}{11} + \frac{30375 \left(\frac{4}{10125 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}\right)^{2}}{44} - \frac{225 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}{11} - \frac{4}{495 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} \right)} + 3 \left(\frac{4}{10125 \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}\right) \log{\left(x - \frac{225 \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}{11} - \frac{6}{11} - \frac{4}{495 \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \frac{30375 \left(\frac{4}{10125 \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}\right)^{2}}{44} \right)}\right) - 4}{15 \left(x^{3} + 3 x + 1\right)}+ \mathrm{constant}$

Respuesta (Indefinida) [src]

  /                                                                                                                               
 |                                  3                 2            /                                /                          2\\
 | / 2          1       \          x       4 - x + 2*x             |       3                        |  6        225*t   30375*t ||
 | |x  + ---------------| dx = C + -- - ----------------- + RootSum|91125*t  - 108*t - 8, t -> t*log|- -- + x - ----- + --------||
 | |                   2|          3             3                 \                                \  11         11       44   //
 | |     / 3          \ |               15 + 15*x  + 45*x                                                                         
 | \     \x  + 3*x + 1/ /                                                                                                         
 |                                                                                                                                
/

$$\int \left(x^{2} + \frac{1}{\left(\left(x^{3} + 3 x\right) + 1\right)^{2}}\right)\, dx = C + \frac{x^{3}}{3} - \frac{2 x^{2} - x + 4}{15 x^{3} + 45 x + 15} + \operatorname{RootSum} {\left(91125 t^{3} - 108 t - 8, \left( t \mapsto t \log{\left(\frac{30375 t^{2}}{44} - \frac{225 t}{11} + x - \frac{6}{11} \right)} \right)\right)}$$

Simplificar

Gráfica

Trazar el gráfico f(x)

Respuesta [src]

 oo                          
  /                          
 |                           
 |  / 2          1       \   
 |  |x  + ---------------| dx
 |  |                   2|   
 |  |     /     3      \ |   
 |  \     \1 + x  + 3*x/ /   
 |                           
/                            
0

$$\int\limits_{0}^{\infty} \left(x^{2} + \frac{1}{\left(x^{3} + 3 x + 1\right)^{2}}\right)\, dx$$

 oo                          
  /                          
 |                           
 |  / 2          1       \   
 |  |x  + ---------------| dx
 |  |                   2|   
 |  |     /     3      \ |   
 |  \     \1 + x  + 3*x/ /   
 |                           
/                            
0

$$\int\limits_{0}^{\infty} \left(x^{2} + \frac{1}{\left(x^{3} + 3 x + 1\right)^{2}}\right)\, dx$$

Integral(x^2 + (1 + x^3 + 3*x)^(-2), (x, 0, oo))

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

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Expresiones idénticas

Expresiones semejantes

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

Límites de integración:

Gráfico:

Definida a trozos:

Solución