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Resolución de integrales/ (5x^4)/ ((5x^4)-4x)/((x^5)-10x+8)

Integral de ((5x^4)-4x)/((x^5)-10x+8) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                 
  /                 
 |                  
 |       4          
 |    5*x  - 4*x    
 |  ------------- dx
 |   5              
 |  x  - 10*x + 8   
 |                  
/                   
0
∫015x4−4x(x5−10x)+8 dx\int\limits_{0}^{1} \frac{5 x^{4} - 4 x}{\left(x^{5} - 10 x\right) + 8}\, dx 
Integral((5*x^4 - 4*x)/(x^5 - 10*x + 8), (x, 0, 1))
Solución detallada

  1. Hay varias maneras de calcular esta integral.

    Método #1

    1. Vuelva a escribir el integrando:

      5x4−4x(x5−10x)+8=x(5x3−4)x5−10x+8\frac{5 x^{4} - 4 x}{\left(x^{5} - 10 x\right) + 8} = \frac{x \left(5 x^{3} - 4\right)}{x^{5} - 10 x + 8}

    2. Vuelva a escribir el integrando:

      x(5x3−4)x5−10x+8=5x4x5−10x+8−4xx5−10x+8\frac{x \left(5 x^{3} - 4\right)}{x^{5} - 10 x + 8} = \frac{5 x^{4}}{x^{5} - 10 x + 8} - \frac{4 x}{x^{5} - 10 x + 8}

    3. Integramos término a término:

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

        ∫5x4x5−10x+8 dx=5∫x4x5−10x+8 dx\int \frac{5 x^{4}}{x^{5} - 10 x + 8}\, dx = 5 \int \frac{x^{4}}{x^{5} - 10 x + 8}\, dx

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

          Pero la integral

          RootSum⁡(400000t5−400000t4+140000t3−18000t2−75t+128,(t↦tlog⁡(−160000t4+120000t3−26000t2+700t+x+204)))\operatorname{RootSum} {\left(400000 t^{5} - 400000 t^{4} + 140000 t^{3} - 18000 t^{2} - 75 t + 128, \left( t \mapsto t \log{\left(- 160000 t^{4} + 120000 t^{3} - 26000 t^{2} + 700 t + x + 204 \right)} \right)\right)}

        Por lo tanto, el resultado es: 5RootSum⁡(400000t5−400000t4+140000t3−18000t2−75t+128,(t↦tlog⁡(−160000t4+120000t3−26000t2+700t+x+204)))5 \operatorname{RootSum} {\left(400000 t^{5} - 400000 t^{4} + 140000 t^{3} - 18000 t^{2} - 75 t + 128, \left( t \mapsto t \log{\left(- 160000 t^{4} + 120000 t^{3} - 26000 t^{2} + 700 t + x + 204 \right)} \right)\right)}

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

        ∫(−4xx5−10x+8) dx=−4∫xx5−10x+8 dx\int \left(- \frac{4 x}{x^{5} - 10 x + 8}\right)\, dx = - 4 \int \frac{x}{x^{5} - 10 x + 8}\, dx

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

          Pero la integral

          RootSum⁡(3200000t5−40000t3−25t−2,(t↦tlog⁡(3200t3−160t2−44t+x+45)))\operatorname{RootSum} {\left(3200000 t^{5} - 40000 t^{3} - 25 t - 2, \left( t \mapsto t \log{\left(3200 t^{3} - 160 t^{2} - 44 t + x + \frac{4}{5} \right)} \right)\right)}

        Por lo tanto, el resultado es: −4RootSum⁡(3200000t5−40000t3−25t−2,(t↦tlog⁡(3200t3−160t2−44t+x+45)))- 4 \operatorname{RootSum} {\left(3200000 t^{5} - 40000 t^{3} - 25 t - 2, \left( t \mapsto t \log{\left(3200 t^{3} - 160 t^{2} - 44 t + x + \frac{4}{5} \right)} \right)\right)}

      El resultado es: −4RootSum⁡(3200000t5−40000t3−25t−2,(t↦tlog⁡(3200t3−160t2−44t+x+45)))+5RootSum⁡(400000t5−400000t4+140000t3−18000t2−75t+128,(t↦tlog⁡(−160000t4+120000t3−26000t2+700t+x+204)))- 4 \operatorname{RootSum} {\left(3200000 t^{5} - 40000 t^{3} - 25 t - 2, \left( t \mapsto t \log{\left(3200 t^{3} - 160 t^{2} - 44 t + x + \frac{4}{5} \right)} \right)\right)} + 5 \operatorname{RootSum} {\left(400000 t^{5} - 400000 t^{4} + 140000 t^{3} - 18000 t^{2} - 75 t + 128, \left( t \mapsto t \log{\left(- 160000 t^{4} + 120000 t^{3} - 26000 t^{2} + 700 t + x + 204 \right)} \right)\right)}

    Método #2

    1. Vuelva a escribir el integrando:

      5x4−4x(x5−10x)+8=5x4(x5−10x)+8−4x(x5−10x)+8\frac{5 x^{4} - 4 x}{\left(x^{5} - 10 x\right) + 8} = \frac{5 x^{4}}{\left(x^{5} - 10 x\right) + 8} - \frac{4 x}{\left(x^{5} - 10 x\right) + 8}

    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:

        ∫5x4(x5−10x)+8 dx=5∫x4(x5−10x)+8 dx\int \frac{5 x^{4}}{\left(x^{5} - 10 x\right) + 8}\, dx = 5 \int \frac{x^{4}}{\left(x^{5} - 10 x\right) + 8}\, dx

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

          Pero la integral

          RootSum⁡(400000t5−400000t4+140000t3−18000t2−75t+128,(t↦tlog⁡(−160000t4+120000t3−26000t2+700t+x+204)))\operatorname{RootSum} {\left(400000 t^{5} - 400000 t^{4} + 140000 t^{3} - 18000 t^{2} - 75 t + 128, \left( t \mapsto t \log{\left(- 160000 t^{4} + 120000 t^{3} - 26000 t^{2} + 700 t + x + 204 \right)} \right)\right)}

        Por lo tanto, el resultado es: 5RootSum⁡(400000t5−400000t4+140000t3−18000t2−75t+128,(t↦tlog⁡(−160000t4+120000t3−26000t2+700t+x+204)))5 \operatorname{RootSum} {\left(400000 t^{5} - 400000 t^{4} + 140000 t^{3} - 18000 t^{2} - 75 t + 128, \left( t \mapsto t \log{\left(- 160000 t^{4} + 120000 t^{3} - 26000 t^{2} + 700 t + x + 204 \right)} \right)\right)}

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

        ∫(−4x(x5−10x)+8) dx=−4∫x(x5−10x)+8 dx\int \left(- \frac{4 x}{\left(x^{5} - 10 x\right) + 8}\right)\, dx = - 4 \int \frac{x}{\left(x^{5} - 10 x\right) + 8}\, dx

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

          Pero la integral

          RootSum⁡(3200000t5−40000t3−25t−2,(t↦tlog⁡(3200t3−160t2−44t+x+45)))\operatorname{RootSum} {\left(3200000 t^{5} - 40000 t^{3} - 25 t - 2, \left( t \mapsto t \log{\left(3200 t^{3} - 160 t^{2} - 44 t + x + \frac{4}{5} \right)} \right)\right)}

        Por lo tanto, el resultado es: −4RootSum⁡(3200000t5−40000t3−25t−2,(t↦tlog⁡(3200t3−160t2−44t+x+45)))- 4 \operatorname{RootSum} {\left(3200000 t^{5} - 40000 t^{3} - 25 t - 2, \left( t \mapsto t \log{\left(3200 t^{3} - 160 t^{2} - 44 t + x + \frac{4}{5} \right)} \right)\right)}

      El resultado es: −4RootSum⁡(3200000t5−40000t3−25t−2,(t↦tlog⁡(3200t3−160t2−44t+x+45)))+5RootSum⁡(400000t5−400000t4+140000t3−18000t2−75t+128,(t↦tlog⁡(−160000t4+120000t3−26000t2+700t+x+204)))- 4 \operatorname{RootSum} {\left(3200000 t^{5} - 40000 t^{3} - 25 t - 2, \left( t \mapsto t \log{\left(3200 t^{3} - 160 t^{2} - 44 t + x + \frac{4}{5} \right)} \right)\right)} + 5 \operatorname{RootSum} {\left(400000 t^{5} - 400000 t^{4} + 140000 t^{3} - 18000 t^{2} - 75 t + 128, \left( t \mapsto t \log{\left(- 160000 t^{4} + 120000 t^{3} - 26000 t^{2} + 700 t + x + 204 \right)} \right)\right)}

  2. Añadimos la constante de integración:

    −4RootSum⁡(3200000t5−40000t3−25t−2,(t↦tlog⁡(3200t3−160t2−44t+x+45)))+5RootSum⁡(400000t5−400000t4+140000t3−18000t2−75t+128,(t↦tlog⁡(−160000t4+120000t3−26000t2+700t+x+204)))+constant- 4 \operatorname{RootSum} {\left(3200000 t^{5} - 40000 t^{3} - 25 t - 2, \left( t \mapsto t \log{\left(3200 t^{3} - 160 t^{2} - 44 t + x + \frac{4}{5} \right)} \right)\right)} + 5 \operatorname{RootSum} {\left(400000 t^{5} - 400000 t^{4} + 140000 t^{3} - 18000 t^{2} - 75 t + 128, \left( t \mapsto t \log{\left(- 160000 t^{4} + 120000 t^{3} - 26000 t^{2} + 700 t + x + 204 \right)} \right)\right)}+ \mathrm{constant}

Respuesta:

−4RootSum⁡(3200000t5−40000t3−25t−2,(t↦tlog⁡(3200t3−160t2−44t+x+45)))+5RootSum⁡(400000t5−400000t4+140000t3−18000t2−75t+128,(t↦tlog⁡(−160000t4+120000t3−26000t2+700t+x+204)))+constant- 4 \operatorname{RootSum} {\left(3200000 t^{5} - 40000 t^{3} - 25 t - 2, \left( t \mapsto t \log{\left(3200 t^{3} - 160 t^{2} - 44 t + x + \frac{4}{5} \right)} \right)\right)} + 5 \operatorname{RootSum} {\left(400000 t^{5} - 400000 t^{4} + 140000 t^{3} - 18000 t^{2} - 75 t + 128, \left( t \mapsto t \log{\left(- 160000 t^{4} + 120000 t^{3} - 26000 t^{2} + 700 t + x + 204 \right)} \right)\right)}+ \mathrm{constant}

Respuesta [src] 
         /                                                                                 /                        2             4                          3\\          /                                                                                 /                        2             4                          3\\
         |        5            4            3            2                                 |  22202756   105045360*t    22560000*t    49727116*t   82747200*t ||          |        5            4            3            2                                 |  22061751   105045360*t    22560000*t    49727116*t   82747200*t ||
- RootSum|400000*t  - 2000000*t  + 3820000*t  - 3370000*t  + 1257325*t - 104744, t -> t*log|- -------- - ------------ - ----------- + ---------- + -----------|| + RootSum|400000*t  - 2000000*t  + 3820000*t  - 3370000*t  + 1257325*t - 104744, t -> t*log|- -------- - ------------ - ----------- + ---------- + -----------||
         \                                                                                 \   141005       28201          28201        28201         28201   //          \                                                                                 \   141005       28201          28201        28201         28201   //
−RootSum⁡(400000t5−2000000t4+3820000t3−3370000t2+1257325t−104744,(t↦tlog⁡(−22560000t428201+82747200t328201−105045360t228201+49727116t28201−22202756141005)))+RootSum⁡(400000t5−2000000t4+3820000t3−3370000t2+1257325t−104744,(t↦tlog⁡(−22560000t428201+82747200t328201−105045360t228201+49727116t28201−22061751141005)))- \operatorname{RootSum} {\left(400000 t^{5} - 2000000 t^{4} + 3820000 t^{3} - 3370000 t^{2} + 1257325 t - 104744, \left( t \mapsto t \log{\left(- \frac{22560000 t^{4}}{28201} + \frac{82747200 t^{3}}{28201} - \frac{105045360 t^{2}}{28201} + \frac{49727116 t}{28201} - \frac{22202756}{141005} \right)} \right)\right)} + \operatorname{RootSum} {\left(400000 t^{5} - 2000000 t^{4} + 3820000 t^{3} - 3370000 t^{2} + 1257325 t - 104744, \left( t \mapsto t \log{\left(- \frac{22560000 t^{4}}{28201} + \frac{82747200 t^{3}}{28201} - \frac{105045360 t^{2}}{28201} + \frac{49727116 t}{28201} - \frac{22061751}{141005} \right)} \right)\right)} 
=
= 
         /                                                                                 /                        2             4                          3\\          /                                                                                 /                        2             4                          3\\
         |        5            4            3            2                                 |  22202756   105045360*t    22560000*t    49727116*t   82747200*t ||          |        5            4            3            2                                 |  22061751   105045360*t    22560000*t    49727116*t   82747200*t ||
- RootSum|400000*t  - 2000000*t  + 3820000*t  - 3370000*t  + 1257325*t - 104744, t -> t*log|- -------- - ------------ - ----------- + ---------- + -----------|| + RootSum|400000*t  - 2000000*t  + 3820000*t  - 3370000*t  + 1257325*t - 104744, t -> t*log|- -------- - ------------ - ----------- + ---------- + -----------||
         \                                                                                 \   141005       28201          28201        28201         28201   //          \                                                                                 \   141005       28201          28201        28201         28201   //
−RootSum⁡(400000t5−2000000t4+3820000t3−3370000t2+1257325t−104744,(t↦tlog⁡(−22560000t428201+82747200t328201−105045360t228201+49727116t28201−22202756141005)))+RootSum⁡(400000t5−2000000t4+3820000t3−3370000t2+1257325t−104744,(t↦tlog⁡(−22560000t428201+82747200t328201−105045360t228201+49727116t28201−22061751141005)))- \operatorname{RootSum} {\left(400000 t^{5} - 2000000 t^{4} + 3820000 t^{3} - 3370000 t^{2} + 1257325 t - 104744, \left( t \mapsto t \log{\left(- \frac{22560000 t^{4}}{28201} + \frac{82747200 t^{3}}{28201} - \frac{105045360 t^{2}}{28201} + \frac{49727116 t}{28201} - \frac{22202756}{141005} \right)} \right)\right)} + \operatorname{RootSum} {\left(400000 t^{5} - 2000000 t^{4} + 3820000 t^{3} - 3370000 t^{2} + 1257325 t - 104744, \left( t \mapsto t \log{\left(- \frac{22560000 t^{4}}{28201} + \frac{82747200 t^{3}}{28201} - \frac{105045360 t^{2}}{28201} + \frac{49727116 t}{28201} - \frac{22061751}{141005} \right)} \right)\right)} 
-RootSum(400000*_t^5 - 2000000*_t^4 + 3820000*_t^3 - 3370000*_t^2 + 1257325*_t - 104744, Lambda(_t, _t*log(-22202756/141005 - 105045360*_t^2/28201 - 22560000*_t^4/28201 + 49727116*_t/28201 + 82747200*_t^3/28201))) + RootSum(400000*_t^5 - 2000000*_t^4 + 3820000*_t^3 - 3370000*_t^2 + 1257325*_t - 104744, Lambda(_t, _t*log(-22061751/141005 - 105045360*_t^2/28201 - 22560000*_t^4/28201 + 49727116*_t/28201 + 82747200*_t^3/28201)))
Respuesta numérica [src] 
-0.308412328324375
-0.308412328324375

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

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