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Resolución de integrales/ tg^2x/ arctg/ (x^5-arctg^2x)/(x^6+7x+4)

Integral de (x^5-arctg^2x)/(x^6+7x+4) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 oo                 
  /                 
 |                  
 |   5       2      
 |  x  - atan (x)   
 |  ------------- dx
 |    6             
 |   x  + 7*x + 4   
 |                  
/                   
1
∫1∞x5−atan⁡2(x)(x6+7x)+4 dx\int\limits_{1}^{\infty} \frac{x^{5} - \operatorname{atan}^{2}{\left(x \right)}}{\left(x^{6} + 7 x\right) + 4}\, dx 
Integral((x^5 - atan(x)^2)/(x^6 + 7*x + 4), (x, 1, oo))
Solución detallada

  1. Vuelva a escribir el integrando:

    x5−atan⁡2(x)(x6+7x)+4=x5(x6+7x)+4−atan⁡2(x)(x6+7x)+4\frac{x^{5} - \operatorname{atan}^{2}{\left(x \right)}}{\left(x^{6} + 7 x\right) + 4} = \frac{x^{5}}{\left(x^{6} + 7 x\right) + 4} - \frac{\operatorname{atan}^{2}{\left(x \right)}}{\left(x^{6} + 7 x\right) + 4}

  2. Integramos término a término:

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

      Pero la integral

      RootSum⁡(319877381t6−319877381t5+127154690t4−24988570t3+2388265t2−80785t−1024,(t↦tlog⁡(199923363125t51792−39984672625t4448+3391710225t3128−1530266905t2448+268452101t1792+x+125047)))\operatorname{RootSum} {\left(319877381 t^{6} - 319877381 t^{5} + 127154690 t^{4} - 24988570 t^{3} + 2388265 t^{2} - 80785 t - 1024, \left( t \mapsto t \log{\left(\frac{199923363125 t^{5}}{1792} - \frac{39984672625 t^{4}}{448} + \frac{3391710225 t^{3}}{128} - \frac{1530266905 t^{2}}{448} + \frac{268452101 t}{1792} + x + \frac{12504}{7} \right)} \right)\right)}

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

      ∫(−atan⁡2(x)(x6+7x)+4) dx=−∫atan⁡2(x)(x6+7x)+4 dx\int \left(- \frac{\operatorname{atan}^{2}{\left(x \right)}}{\left(x^{6} + 7 x\right) + 4}\right)\, dx = - \int \frac{\operatorname{atan}^{2}{\left(x \right)}}{\left(x^{6} + 7 x\right) + 4}\, dx

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

        Pero la integral

        ∫atan⁡2(x)x6+7x+4 dx\int \frac{\operatorname{atan}^{2}{\left(x \right)}}{x^{6} + 7 x + 4}\, dx

      Por lo tanto, el resultado es: −∫atan⁡2(x)x6+7x+4 dx- \int \frac{\operatorname{atan}^{2}{\left(x \right)}}{x^{6} + 7 x + 4}\, dx

    El resultado es: −∫atan⁡2(x)x6+7x+4 dx+RootSum⁡(319877381t6−319877381t5+127154690t4−24988570t3+2388265t2−80785t−1024,(t↦tlog⁡(199923363125t51792−39984672625t4448+3391710225t3128−1530266905t2448+268452101t1792+x+125047)))- \int \frac{\operatorname{atan}^{2}{\left(x \right)}}{x^{6} + 7 x + 4}\, dx + \operatorname{RootSum} {\left(319877381 t^{6} - 319877381 t^{5} + 127154690 t^{4} - 24988570 t^{3} + 2388265 t^{2} - 80785 t - 1024, \left( t \mapsto t \log{\left(\frac{199923363125 t^{5}}{1792} - \frac{39984672625 t^{4}}{448} + \frac{3391710225 t^{3}}{128} - \frac{1530266905 t^{2}}{448} + \frac{268452101 t}{1792} + x + \frac{12504}{7} \right)} \right)\right)}

  3. Añadimos la constante de integración:

    −∫atan⁡2(x)x6+7x+4 dx+RootSum⁡(319877381t6−319877381t5+127154690t4−24988570t3+2388265t2−80785t−1024,(t↦tlog⁡(199923363125t51792−39984672625t4448+3391710225t3128−1530266905t2448+268452101t1792+x+125047)))+constant- \int \frac{\operatorname{atan}^{2}{\left(x \right)}}{x^{6} + 7 x + 4}\, dx + \operatorname{RootSum} {\left(319877381 t^{6} - 319877381 t^{5} + 127154690 t^{4} - 24988570 t^{3} + 2388265 t^{2} - 80785 t - 1024, \left( t \mapsto t \log{\left(\frac{199923363125 t^{5}}{1792} - \frac{39984672625 t^{4}}{448} + \frac{3391710225 t^{3}}{128} - \frac{1530266905 t^{2}}{448} + \frac{268452101 t}{1792} + x + \frac{12504}{7} \right)} \right)\right)}+ \mathrm{constant}

Respuesta:

−∫atan⁡2(x)x6+7x+4 dx+RootSum⁡(319877381t6−319877381t5+127154690t4−24988570t3+2388265t2−80785t−1024,(t↦tlog⁡(199923363125t51792−39984672625t4448+3391710225t3128−1530266905t2448+268452101t1792+x+125047)))+constant- \int \frac{\operatorname{atan}^{2}{\left(x \right)}}{x^{6} + 7 x + 4}\, dx + \operatorname{RootSum} {\left(319877381 t^{6} - 319877381 t^{5} + 127154690 t^{4} - 24988570 t^{3} + 2388265 t^{2} - 80785 t - 1024, \left( t \mapsto t \log{\left(\frac{199923363125 t^{5}}{1792} - \frac{39984672625 t^{4}}{448} + \frac{3391710225 t^{3}}{128} - \frac{1530266905 t^{2}}{448} + \frac{268452101 t}{1792} + x + \frac{12504}{7} \right)} \right)\right)}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
  /                         /                                                                                                                                                                                                                         
 |                         |                                                                                                                                                                                                                          
 |  5       2              |       2                  /                                                                                                  /                         4               2                             3                 5\\
 | x  - atan (x)           |   atan (x)               |           6              5              4             3            2                             |12504       39984672625*t    1530266905*t    268452101*t   3391710225*t    199923363125*t ||
 | ------------- dx = C -  | ------------ dx + RootSum|319877381*t  - 319877381*t  + 127154690*t  - 24988570*t  + 2388265*t  - 80785*t - 1024, t -> t*log|----- + x - -------------- - ------------- + ----------- + ------------- + ---------------||
 |   6                     |      6                   \                                                                                                  \  7              448              448            1792           128              1792     //
 |  x  + 7*x + 4           | 4 + x  + 7*x                                                                                                                                                                                                             
 |                         |                                                                                                                                                                                                                          
/                         /
∫x5−atan⁡2(x)(x6+7x)+4 dx=C−∫atan⁡2(x)x6+7x+4 dx+RootSum⁡(319877381t6−319877381t5+127154690t4−24988570t3+2388265t2−80785t−1024,(t↦tlog⁡(199923363125t51792−39984672625t4448+3391710225t3128−1530266905t2448+268452101t1792+x+125047)))\int \frac{x^{5} - \operatorname{atan}^{2}{\left(x \right)}}{\left(x^{6} + 7 x\right) + 4}\, dx = C - \int \frac{\operatorname{atan}^{2}{\left(x \right)}}{x^{6} + 7 x + 4}\, dx + \operatorname{RootSum} {\left(319877381 t^{6} - 319877381 t^{5} + 127154690 t^{4} - 24988570 t^{3} + 2388265 t^{2} - 80785 t - 1024, \left( t \mapsto t \log{\left(\frac{199923363125 t^{5}}{1792} - \frac{39984672625 t^{4}}{448} + \frac{3391710225 t^{3}}{128} - \frac{1530266905 t^{2}}{448} + \frac{268452101 t}{1792} + x + \frac{12504}{7} \right)} \right)\right)}
Simplificar
Respuesta [src] 
 oo                 
  /                 
 |                  
 |   5       2      
 |  x  - atan (x)   
 |  ------------- dx
 |        6         
 |   4 + x  + 7*x   
 |                  
/                   
1
∫1∞x5−atan⁡2(x)x6+7x+4 dx\int\limits_{1}^{\infty} \frac{x^{5} - \operatorname{atan}^{2}{\left(x \right)}}{x^{6} + 7 x + 4}\, dx 
=
= 
 oo                 
  /                 
 |                  
 |   5       2      
 |  x  - atan (x)   
 |  ------------- dx
 |        6         
 |   4 + x  + 7*x   
 |                  
/                   
1
∫1∞x5−atan⁡2(x)x6+7x+4 dx\int\limits_{1}^{\infty} \frac{x^{5} - \operatorname{atan}^{2}{\left(x \right)}}{x^{6} + 7 x + 4}\, dx 
Integral((x^5 - atan(x)^2)/(4 + x^6 + 7*x), (x, 1, oo))

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

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