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Resolución de integrales/ x^2+2/ 4-x^2/ x^4-x/ x^2+4/ (x^2+4)/(x^4-x^2+2)

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

Solución

Ha introducido [src]

 oo               
  /               
 |                
 |      2         
 |     x  + 4     
 |  ----------- dx
 |   4    2       
 |  x  - x  + 2   
 |                
/                 
0

$$\int\limits_{0}^{\infty} \frac{x^{2} + 4}{\left(x^{4} - x^{2}\right) + 2}\, dx$$

Integral((x^2 + 4)/(x^4 - x^2 + 2), (x, 0, oo))

Solución detallada

Vuelva a escribir el integrando:

$\frac{x^{2} + 4}{\left(x^{4} - x^{2}\right) + 2} = \frac{x^{2}}{\left(x^{4} - x^{2}\right) + 2} + \frac{4}{\left(x^{4} - x^{2}\right) + 2}$
Integramos término a término:
1. No puedo encontrar los pasos en la búsqueda de esta integral.
  
  Pero la integral
  
  $\operatorname{RootSum} {\left(392 t^{4} + 14 t^{2} + 1, \left( t \mapsto t \log{\left(112 t^{3} + 2 t + x \right)} \right)\right)}$
1. La integral del producto de una función por una constante es la constante por la integral de esta función:
  
  $\int \frac{4}{\left(x^{4} - x^{2}\right) + 2}\, dx = 4 \int \frac{1}{\left(x^{4} - x^{2}\right) + 2}\, dx$
  1. No puedo encontrar los pasos en la búsqueda de esta integral.
    
    Pero la integral
    
    $\operatorname{RootSum} {\left(1568 t^{4} + 28 t^{2} + 1, \left( t \mapsto t \log{\left(- 112 t^{3} + 6 t + x \right)} \right)\right)}$
  Por lo tanto, el resultado es: $4 \operatorname{RootSum} {\left(1568 t^{4} + 28 t^{2} + 1, \left( t \mapsto t \log{\left(- 112 t^{3} + 6 t + x \right)} \right)\right)}$
El resultado es: $4 \operatorname{RootSum} {\left(1568 t^{4} + 28 t^{2} + 1, \left( t \mapsto t \log{\left(- 112 t^{3} + 6 t + x \right)} \right)\right)} + \operatorname{RootSum} {\left(392 t^{4} + 14 t^{2} + 1, \left( t \mapsto t \log{\left(112 t^{3} + 2 t + x \right)} \right)\right)}$
Ahora simplificar:

$\left(- \frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14}\right) \log{\left(x - \frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{7} - \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{7} + 112 \left(- \frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14}\right)^{3} \right)} + \left(- \frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} + \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14}\right) \log{\left(x - \frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{7} + 112 \left(- \frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} + \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14}\right)^{3} + \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{7} \right)} + \left(\frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14}\right) \log{\left(x + \frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{7} - \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{7} + 112 \left(\frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14}\right)^{3} \right)} + \left(\frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} + \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14}\right) \log{\left(x + \frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{7} + 112 \left(\frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} + \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14}\right)^{3} + \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{7} \right)} + 4 \left(- \frac{2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28} - \frac{2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28}\right) \log{\left(x - \frac{3 \cdot 2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - \frac{3 \cdot 2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - 112 \left(- \frac{2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28} - \frac{2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28}\right)^{3} \right)} + 4 \left(- \frac{2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28} + \frac{2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28}\right) \log{\left(x - \frac{3 \cdot 2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - 112 \left(- \frac{2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28} + \frac{2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28}\right)^{3} + \frac{3 \cdot 2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} \right)} + 4 \left(\frac{2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28} - \frac{2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28}\right) \log{\left(x + \frac{3 \cdot 2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - \frac{3 \cdot 2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - 112 \left(\frac{2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28} - \frac{2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28}\right)^{3} \right)} + 4 \left(\frac{2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28} + \frac{2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28}\right) \log{\left(x + \frac{3 \cdot 2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - 112 \left(\frac{2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28} + \frac{2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28}\right)^{3} + \frac{3 \cdot 2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} \right)}$
Añadimos la constante de integración:

$\left(- \frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14}\right) \log{\left(x - \frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{7} - \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{7} + 112 \left(- \frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14}\right)^{3} \right)} + \left(- \frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} + \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14}\right) \log{\left(x - \frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{7} + 112 \left(- \frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} + \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14}\right)^{3} + \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{7} \right)} + \left(\frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14}\right) \log{\left(x + \frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{7} - \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{7} + 112 \left(\frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14}\right)^{3} \right)} + \left(\frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} + \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14}\right) \log{\left(x + \frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{7} + 112 \left(\frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} + \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14}\right)^{3} + \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{7} \right)} + 4 \left(- \frac{2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28} - \frac{2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28}\right) \log{\left(x - \frac{3 \cdot 2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - \frac{3 \cdot 2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - 112 \left(- \frac{2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28} - \frac{2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28}\right)^{3} \right)} + 4 \left(- \frac{2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28} + \frac{2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28}\right) \log{\left(x - \frac{3 \cdot 2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - 112 \left(- \frac{2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28} + \frac{2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28}\right)^{3} + \frac{3 \cdot 2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} \right)} + 4 \left(\frac{2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28} - \frac{2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28}\right) \log{\left(x + \frac{3 \cdot 2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - \frac{3 \cdot 2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - 112 \left(\frac{2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28} - \frac{2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28}\right)^{3} \right)} + 4 \left(\frac{2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28} + \frac{2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28}\right) \log{\left(x + \frac{3 \cdot 2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - 112 \left(\frac{2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28} + \frac{2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28}\right)^{3} + \frac{3 \cdot 2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} \right)}+ \mathrm{constant}$

Respuesta:

$\left(- \frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14}\right) \log{\left(x - \frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{7} - \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{7} + 112 \left(- \frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14}\right)^{3} \right)} + \left(- \frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} + \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14}\right) \log{\left(x - \frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{7} + 112 \left(- \frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} + \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14}\right)^{3} + \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{7} \right)} + \left(\frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14}\right) \log{\left(x + \frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{7} - \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{7} + 112 \left(\frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14}\right)^{3} \right)} + \left(\frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} + \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14}\right) \log{\left(x + \frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{7} + 112 \left(\frac{\sqrt[4]{2} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} + \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14}\right)^{3} + \frac{\sqrt[4]{2} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{7} \right)} + 4 \left(- \frac{2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28} - \frac{2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28}\right) \log{\left(x - \frac{3 \cdot 2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - \frac{3 \cdot 2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - 112 \left(- \frac{2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28} - \frac{2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28}\right)^{3} \right)} + 4 \left(- \frac{2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28} + \frac{2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28}\right) \log{\left(x - \frac{3 \cdot 2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - 112 \left(- \frac{2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28} + \frac{2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28}\right)^{3} + \frac{3 \cdot 2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} \right)} + 4 \left(\frac{2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28} - \frac{2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28}\right) \log{\left(x + \frac{3 \cdot 2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - \frac{3 \cdot 2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - 112 \left(\frac{2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28} - \frac{2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28}\right)^{3} \right)} + 4 \left(\frac{2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28} + \frac{2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28}\right) \log{\left(x + \frac{3 \cdot 2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} - 112 \left(\frac{2^{\frac{3}{4}} \sqrt{7} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28} + \frac{2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{28}\right)^{3} + \frac{3 \cdot 2^{\frac{3}{4}} \sqrt{7} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{14} \right)}+ \mathrm{constant}$

Respuesta (Indefinida) [src]

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 |     2                                                                                                                                        
 |    x  + 4                     /      4       2                /         3      \\          /     4       2                /               3\\
 | ----------- dx = C + 4*RootSum\1568*t  + 28*t  + 1, t -> t*log\x - 112*t  + 6*t// + RootSum\392*t  + 14*t  + 1, t -> t*log\x + 2*t + 112*t //
 |  4    2                                                                                                                                      
 | x  - x  + 2                                                                                                                                  
 |                                                                                                                                              
/

$$\int \frac{x^{2} + 4}{\left(x^{4} - x^{2}\right) + 2}\, dx = C + \operatorname{RootSum} {\left(392 t^{4} + 14 t^{2} + 1, \left( t \mapsto t \log{\left(112 t^{3} + 2 t + x \right)} \right)\right)} + 4 \operatorname{RootSum} {\left(1568 t^{4} + 28 t^{2} + 1, \left( t \mapsto t \log{\left(- 112 t^{3} + 6 t + x \right)} \right)\right)}$$

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Trazar el gráfico f(x)

Respuesta [src]

 oo               
  /               
 |                
 |          2     
 |     4 + x      
 |  ----------- dx
 |       4    2   
 |  2 + x  - x    
 |                
/                 
0

$$\int\limits_{0}^{\infty} \frac{x^{2} + 4}{x^{4} - x^{2} + 2}\, dx$$

 oo               
  /               
 |                
 |          2     
 |     4 + x      
 |  ----------- dx
 |       4    2   
 |  2 + x  - x    
 |                
/                 
0

$$\int\limits_{0}^{\infty} \frac{x^{2} + 4}{x^{4} - x^{2} + 2}\, dx$$

Integral((4 + x^2)/(2 + x^4 - x^2), (x, 0, oo))

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

Límites de integración:

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Solución