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Resolución de integrales/ x^2+1/ (x^2+1)/(2x^5+x+18)

Integral de (x^2+1)/(2x^5+x+18) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 oo                 
  /                 
 |                  
 |       2          
 |      x  + 1      
 |  ------------- dx
 |     5            
 |  2*x  + x + 18   
 |                  
/                   
1
∫1∞x2+1(2x5+x)+18 dx\int\limits_{1}^{\infty} \frac{x^{2} + 1}{\left(2 x^{5} + x\right) + 18}\, dx 
Integral((x^2 + 1)/(2*x^5 + x + 18), (x, 1, oo))
Solución detallada

  1. Vuelva a escribir el integrando:

    x2+1(2x5+x)+18=x2(2x5+x)+18+1(2x5+x)+18\frac{x^{2} + 1}{\left(2 x^{5} + x\right) + 18} = \frac{x^{2}}{\left(2 x^{5} + x\right) + 18} + \frac{1}{\left(2 x^{5} + x\right) + 18}

  2. Integramos término a término:

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

      Pero la integral

      RootSum⁡(2624401024t5+64t3−16200t2+t−324,(t↦tlog⁡(−40356320620501536000t41494678229396907+85196679052504096t340356312193716489+1345209744206221200t21494678229396907+4484034968400128t40356312193716489+x−108666572101494678229396907)))\operatorname{RootSum} {\left(2624401024 t^{5} + 64 t^{3} - 16200 t^{2} + t - 324, \left( t \mapsto t \log{\left(- \frac{40356320620501536000 t^{4}}{1494678229396907} + \frac{85196679052504096 t^{3}}{40356312193716489} + \frac{1345209744206221200 t^{2}}{1494678229396907} + \frac{4484034968400128 t}{40356312193716489} + x - \frac{10866657210}{1494678229396907} \right)} \right)\right)}

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

      Pero la integral

      RootSum⁡(656100256t5−160t3−80t2−15t−1,(t↦tlog⁡(−2624401024t4455625+656100256t3455625−18224936t250625+41006426t455625+x+16455625)))\operatorname{RootSum} {\left(656100256 t^{5} - 160 t^{3} - 80 t^{2} - 15 t - 1, \left( t \mapsto t \log{\left(- \frac{2624401024 t^{4}}{455625} + \frac{656100256 t^{3}}{455625} - \frac{18224936 t^{2}}{50625} + \frac{41006426 t}{455625} + x + \frac{16}{455625} \right)} \right)\right)}

    El resultado es: RootSum⁡(2624401024t5+64t3−16200t2+t−324,(t↦tlog⁡(−40356320620501536000t41494678229396907+85196679052504096t340356312193716489+1345209744206221200t21494678229396907+4484034968400128t40356312193716489+x−108666572101494678229396907)))+RootSum⁡(656100256t5−160t3−80t2−15t−1,(t↦tlog⁡(−2624401024t4455625+656100256t3455625−18224936t250625+41006426t455625+x+16455625)))\operatorname{RootSum} {\left(2624401024 t^{5} + 64 t^{3} - 16200 t^{2} + t - 324, \left( t \mapsto t \log{\left(- \frac{40356320620501536000 t^{4}}{1494678229396907} + \frac{85196679052504096 t^{3}}{40356312193716489} + \frac{1345209744206221200 t^{2}}{1494678229396907} + \frac{4484034968400128 t}{40356312193716489} + x - \frac{10866657210}{1494678229396907} \right)} \right)\right)} + \operatorname{RootSum} {\left(656100256 t^{5} - 160 t^{3} - 80 t^{2} - 15 t - 1, \left( t \mapsto t \log{\left(- \frac{2624401024 t^{4}}{455625} + \frac{656100256 t^{3}}{455625} - \frac{18224936 t^{2}}{50625} + \frac{41006426 t}{455625} + x + \frac{16}{455625} \right)} \right)\right)}

  3. Añadimos la constante de integración:

    RootSum⁡(2624401024t5+64t3−16200t2+t−324,(t↦tlog⁡(−40356320620501536000t41494678229396907+85196679052504096t340356312193716489+1345209744206221200t21494678229396907+4484034968400128t40356312193716489+x−108666572101494678229396907)))+RootSum⁡(656100256t5−160t3−80t2−15t−1,(t↦tlog⁡(−2624401024t4455625+656100256t3455625−18224936t250625+41006426t455625+x+16455625)))+constant\operatorname{RootSum} {\left(2624401024 t^{5} + 64 t^{3} - 16200 t^{2} + t - 324, \left( t \mapsto t \log{\left(- \frac{40356320620501536000 t^{4}}{1494678229396907} + \frac{85196679052504096 t^{3}}{40356312193716489} + \frac{1345209744206221200 t^{2}}{1494678229396907} + \frac{4484034968400128 t}{40356312193716489} + x - \frac{10866657210}{1494678229396907} \right)} \right)\right)} + \operatorname{RootSum} {\left(656100256 t^{5} - 160 t^{3} - 80 t^{2} - 15 t - 1, \left( t \mapsto t \log{\left(- \frac{2624401024 t^{4}}{455625} + \frac{656100256 t^{3}}{455625} - \frac{18224936 t^{2}}{50625} + \frac{41006426 t}{455625} + x + \frac{16}{455625} \right)} \right)\right)}+ \mathrm{constant}

Respuesta:

RootSum⁡(2624401024t5+64t3−16200t2+t−324,(t↦tlog⁡(−40356320620501536000t41494678229396907+85196679052504096t340356312193716489+1345209744206221200t21494678229396907+4484034968400128t40356312193716489+x−108666572101494678229396907)))+RootSum⁡(656100256t5−160t3−80t2−15t−1,(t↦tlog⁡(−2624401024t4455625+656100256t3455625−18224936t250625+41006426t455625+x+16455625)))+constant\operatorname{RootSum} {\left(2624401024 t^{5} + 64 t^{3} - 16200 t^{2} + t - 324, \left( t \mapsto t \log{\left(- \frac{40356320620501536000 t^{4}}{1494678229396907} + \frac{85196679052504096 t^{3}}{40356312193716489} + \frac{1345209744206221200 t^{2}}{1494678229396907} + \frac{4484034968400128 t}{40356312193716489} + x - \frac{10866657210}{1494678229396907} \right)} \right)\right)} + \operatorname{RootSum} {\left(656100256 t^{5} - 160 t^{3} - 80 t^{2} - 15 t - 1, \left( t \mapsto t \log{\left(- \frac{2624401024 t^{4}}{455625} + \frac{656100256 t^{3}}{455625} - \frac{18224936 t^{2}}{50625} + \frac{41006426 t}{455625} + x + \frac{16}{455625} \right)} \right)\right)}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
  /                                                                                                                                                                                                                                                                                                                                                   
 |                                                                                                                                                                                                                                                                                                                                                    
 |      2                        /                                                    /                         4             2                           3\\          /                                                      /                                               4                                           3                        2\\
 |     x  + 1                    |           5        3       2                       |  16         2624401024*t    18224936*t    41006426*t   656100256*t ||          |            5       3          2                      |    10866657210          40356320620501536000*t    4484034968400128*t   85196679052504096*t    1345209744206221200*t ||
 | ------------- dx = C + RootSum|656100256*t  - 160*t  - 80*t  - 15*t - 1, t -> t*log|------ + x - ------------- - ----------- + ---------- + ------------|| + RootSum|2624401024*t  + 64*t  - 16200*t  + t - 324, t -> t*log|- ---------------- + x - ----------------------- + ------------------ + -------------------- + ----------------------||
 |    5                          \                                                    \455625           455625         50625        455625        455625   //          \                                                      \  1494678229396907           1494678229396907      40356312193716489     40356312193716489        1494678229396907   //
 | 2*x  + x + 18                                                                                                                                                                                                                                                                                                                                      
 |                                                                                                                                                                                                                                                                                                                                                    
/
∫x2+1(2x5+x)+18 dx=C+RootSum⁡(2624401024t5+64t3−16200t2+t−324,(t↦tlog⁡(−40356320620501536000t41494678229396907+85196679052504096t340356312193716489+1345209744206221200t21494678229396907+4484034968400128t40356312193716489+x−108666572101494678229396907)))+RootSum⁡(656100256t5−160t3−80t2−15t−1,(t↦tlog⁡(−2624401024t4455625+656100256t3455625−18224936t250625+41006426t455625+x+16455625)))\int \frac{x^{2} + 1}{\left(2 x^{5} + x\right) + 18}\, dx = C + \operatorname{RootSum} {\left(2624401024 t^{5} + 64 t^{3} - 16200 t^{2} + t - 324, \left( t \mapsto t \log{\left(- \frac{40356320620501536000 t^{4}}{1494678229396907} + \frac{85196679052504096 t^{3}}{40356312193716489} + \frac{1345209744206221200 t^{2}}{1494678229396907} + \frac{4484034968400128 t}{40356312193716489} + x - \frac{10866657210}{1494678229396907} \right)} \right)\right)} + \operatorname{RootSum} {\left(656100256 t^{5} - 160 t^{3} - 80 t^{2} - 15 t - 1, \left( t \mapsto t \log{\left(- \frac{2624401024 t^{4}}{455625} + \frac{656100256 t^{3}}{455625} - \frac{18224936 t^{2}}{50625} + \frac{41006426 t}{455625} + x + \frac{16}{455625} \right)} \right)\right)}
Simplificar
Gráfica
1.00001.01001.00101.00201.00301.00401.00501.00601.00701.00801.00900.25-0.25
Trazar el gráfico f(x)
Respuesta [src] 
 oo                 
  /                 
 |                  
 |           2      
 |      1 + x       
 |  ------------- dx
 |              5   
 |  18 + x + 2*x    
 |                  
/                   
1
∫1∞x2+12x5+x+18 dx\int\limits_{1}^{\infty} \frac{x^{2} + 1}{2 x^{5} + x + 18}\, dx 
=
= 
 oo                 
  /                 
 |                  
 |           2      
 |      1 + x       
 |  ------------- dx
 |              5   
 |  18 + x + 2*x    
 |                  
/                   
1
∫1∞x2+12x5+x+18 dx\int\limits_{1}^{\infty} \frac{x^{2} + 1}{2 x^{5} + x + 18}\, dx 
Integral((1 + x^2)/(18 + x + 2*x^5), (x, 1, oo))

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

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