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Resolución de integrales/ x^3+3/ 3*x+1/ x^2/(x^3+3*x+1)^3

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                   
  /                   
 |                    
 |          2         
 |         x          
 |  --------------- dx
 |                3   
 |  / 3          \    
 |  \x  + 3*x + 1/    
 |                    
/                     
0
$$\int\limits_{0}^{1} \frac{x^{2}}{\left(\left(x^{3} + 3 x\right) + 1\right)^{3}}\, dx$$ 
Integral(x^2/(x^3 + 3*x + 1)^3, (x, 0, 1))
Respuesta (Indefinida) [src] 
                                                                                                                                                              /                                                                                               2                                                                                 \
                                                                                                                                                              |                      /     _________________________                                         \                                                                                  |
                                                                                                                                                              |                      |    /                     ___                                          |                                                                                  |
                                                                                                                                                              |                      |   /     131       7379*\/ 5                       59                  |                                                                                  |
                                                                                                                                                              |             29615625*|3 /   --------- + -----------  - --------------------------------------|                                                                                  |
                                                                                                                                                              |                      |\/    123018750   15377343750                 _________________________|                                                         _________________________|
  /                                                                                                                                                           |                      |                                             /                     ___ |                                                        /                     ___ |
 |                                                                               /     _________________________                                         \    |                      |                                            /     131       7379*\/ 5  |                                                       /     131       7379*\/ 5  |
 |         2                             4       5       2       3               |    /                     ___                                          |    |                      |                                 2278125*3 /   --------- + ----------- |                                             681750*3 /   --------- + ----------- |
 |        x                     21 + 11*x  + 18*x  + 81*x  + 90*x  + 104*x       |   /     131       7379*\/ 5                       59                  |    |  1534                \                                         \/    123018750   15377343750 /                    11918                           \/    123018750   15377343750 |
 | --------------- dx = C + -------------------------------------------------- + |3 /   --------- + -----------  - --------------------------------------|*log|- ---- + x - ----------------------------------------------------------------------------------- - -------------------------------------- + -------------------------------------|
 |               3                     6        3                  4         2   |\/    123018750   15377343750                 _________________________|    |  7379                                               7379                                                       _________________________                    7379                |
 | / 3          \           450 + 450*x  + 900*x  + 2700*x + 2700*x  + 4050*x    |                                             /                     ___ |    |                                                                                                               /                     ___                                         |
 | \x  + 3*x + 1/                                                                |                                            /     131       7379*\/ 5  |    |                                                                                                              /     131       7379*\/ 5                                          |
 |                                                                               |                                 2278125*3 /   --------- + ----------- |    |                                                                                                   4980825*3 /   --------- + -----------                                         |
/                                                                                \                                         \/    123018750   15377343750 /    \                                                                                                           \/    123018750   15377343750                                         /
$$\int \frac{x^{2}}{\left(\left(x^{3} + 3 x\right) + 1\right)^{3}}\, dx = C + \frac{18 x^{5} + 11 x^{4} + 90 x^{3} + 81 x^{2} + 104 x + 21}{450 x^{6} + 2700 x^{4} + 900 x^{3} + 4050 x^{2} + 2700 x + 450} + \left(- \frac{59}{2278125 \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}} + \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}\right) \log{\left(x - \frac{29615625 \left(- \frac{59}{2278125 \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}} + \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}\right)^{2}}{7379} - \frac{1534}{7379} - \frac{11918}{4980825 \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}} + \frac{681750 \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}}{7379} \right)}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
                                                                                     /                                                                                         2                                                                                 \                                                                                /                                                                                           2                                                                                 \
                                                                                     |                /     _________________________                                         \                                                                                  |                                                                                |                  /     _________________________                                         \                                                                                  |
                                                                                     |                |    /                     ___                                          |                                                                                  |                                                                                |                  |    /                     ___                                          |                                                                                  |
                                                                                     |                |   /     131       7379*\/ 5                       59                  |                                                                                  |                                                                                |                  |   /     131       7379*\/ 5                       59                  |                                                                                  |
                                                                                     |       29615625*|3 /   --------- + -----------  - --------------------------------------|                                                                                  |                                                                                |         29615625*|3 /   --------- + -----------  - --------------------------------------|                                                                                  |
                                                                                     |                |\/    123018750   15377343750                 _________________________|                                                         _________________________|                                                                                |                  |\/    123018750   15377343750                 _________________________|                                                         _________________________|
                                                                                     |                |                                             /                     ___ |                                                        /                     ___ |                                                                                |                  |                                             /                     ___ |                                                        /                     ___ |
        /     _________________________                                         \    |                |                                            /     131       7379*\/ 5  |                                                       /     131       7379*\/ 5  |   /     _________________________                                         \    |                  |                                            /     131       7379*\/ 5  |                                                       /     131       7379*\/ 5  |
        |    /                     ___                                          |    |                |                                 2278125*3 /   --------- + ----------- |                                             681750*3 /   --------- + ----------- |   |    /                     ___                                          |    |                  |                                 2278125*3 /   --------- + ----------- |                                             681750*3 /   --------- + ----------- |
   4    |   /     131       7379*\/ 5                       59                  |    |5845            \                                         \/    123018750   15377343750 /                    11918                           \/    123018750   15377343750 |   |   /     131       7379*\/ 5                       59                  |    |  1534            \                                         \/    123018750   15377343750 /                    11918                           \/    123018750   15377343750 |
- --- + |3 /   --------- + -----------  - --------------------------------------|*log|---- - ----------------------------------------------------------------------------------- - -------------------------------------- + -------------------------------------| - |3 /   --------- + -----------  - --------------------------------------|*log|- ---- - ----------------------------------------------------------------------------------- - -------------------------------------- + -------------------------------------|
  225   |\/    123018750   15377343750                 _________________________|    |7379                                           7379                                                       _________________________                    7379                |   |\/    123018750   15377343750                 _________________________|    |  7379                                           7379                                                       _________________________                    7379                |
        |                                             /                     ___ |    |                                                                                                         /                     ___                                         |   |                                             /                     ___ |    |                                                                                                           /                     ___                                         |
        |                                            /     131       7379*\/ 5  |    |                                                                                                        /     131       7379*\/ 5                                          |   |                                            /     131       7379*\/ 5  |    |                                                                                                          /     131       7379*\/ 5                                          |
        |                                 2278125*3 /   --------- + ----------- |    |                                                                                             4980825*3 /   --------- + -----------                                         |   |                                 2278125*3 /   --------- + ----------- |    |                                                                                               4980825*3 /   --------- + -----------                                         |
        \                                         \/    123018750   15377343750 /    \                                                                                                     \/    123018750   15377343750                                         /   \                                         \/    123018750   15377343750 /    \                                                                                                       \/    123018750   15377343750                                         /
$$- \frac{4}{225} + \left(- \frac{59}{2278125 \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}} + \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}\right) \log{\left(- \frac{29615625 \left(- \frac{59}{2278125 \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}} + \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}\right)^{2}}{7379} - \frac{11918}{4980825 \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}} + \frac{5845}{7379} + \frac{681750 \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}}{7379} \right)} - \left(- \frac{59}{2278125 \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}} + \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}\right) \log{\left(- \frac{29615625 \left(- \frac{59}{2278125 \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}} + \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}\right)^{2}}{7379} - \frac{1534}{7379} - \frac{11918}{4980825 \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}} + \frac{681750 \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}}{7379} \right)}$$ 
=
= 
                                                                                     /                                                                                         2                                                                                 \                                                                                /                                                                                           2                                                                                 \
                                                                                     |                /     _________________________                                         \                                                                                  |                                                                                |                  /     _________________________                                         \                                                                                  |
                                                                                     |                |    /                     ___                                          |                                                                                  |                                                                                |                  |    /                     ___                                          |                                                                                  |
                                                                                     |                |   /     131       7379*\/ 5                       59                  |                                                                                  |                                                                                |                  |   /     131       7379*\/ 5                       59                  |                                                                                  |
                                                                                     |       29615625*|3 /   --------- + -----------  - --------------------------------------|                                                                                  |                                                                                |         29615625*|3 /   --------- + -----------  - --------------------------------------|                                                                                  |
                                                                                     |                |\/    123018750   15377343750                 _________________________|                                                         _________________________|                                                                                |                  |\/    123018750   15377343750                 _________________________|                                                         _________________________|
                                                                                     |                |                                             /                     ___ |                                                        /                     ___ |                                                                                |                  |                                             /                     ___ |                                                        /                     ___ |
        /     _________________________                                         \    |                |                                            /     131       7379*\/ 5  |                                                       /     131       7379*\/ 5  |   /     _________________________                                         \    |                  |                                            /     131       7379*\/ 5  |                                                       /     131       7379*\/ 5  |
        |    /                     ___                                          |    |                |                                 2278125*3 /   --------- + ----------- |                                             681750*3 /   --------- + ----------- |   |    /                     ___                                          |    |                  |                                 2278125*3 /   --------- + ----------- |                                             681750*3 /   --------- + ----------- |
   4    |   /     131       7379*\/ 5                       59                  |    |5845            \                                         \/    123018750   15377343750 /                    11918                           \/    123018750   15377343750 |   |   /     131       7379*\/ 5                       59                  |    |  1534            \                                         \/    123018750   15377343750 /                    11918                           \/    123018750   15377343750 |
- --- + |3 /   --------- + -----------  - --------------------------------------|*log|---- - ----------------------------------------------------------------------------------- - -------------------------------------- + -------------------------------------| - |3 /   --------- + -----------  - --------------------------------------|*log|- ---- - ----------------------------------------------------------------------------------- - -------------------------------------- + -------------------------------------|
  225   |\/    123018750   15377343750                 _________________________|    |7379                                           7379                                                       _________________________                    7379                |   |\/    123018750   15377343750                 _________________________|    |  7379                                           7379                                                       _________________________                    7379                |
        |                                             /                     ___ |    |                                                                                                         /                     ___                                         |   |                                             /                     ___ |    |                                                                                                           /                     ___                                         |
        |                                            /     131       7379*\/ 5  |    |                                                                                                        /     131       7379*\/ 5                                          |   |                                            /     131       7379*\/ 5  |    |                                                                                                          /     131       7379*\/ 5                                          |
        |                                 2278125*3 /   --------- + ----------- |    |                                                                                             4980825*3 /   --------- + -----------                                         |   |                                 2278125*3 /   --------- + ----------- |    |                                                                                               4980825*3 /   --------- + -----------                                         |
        \                                         \/    123018750   15377343750 /    \                                                                                                     \/    123018750   15377343750                                         /   \                                         \/    123018750   15377343750 /    \                                                                                                       \/    123018750   15377343750                                         /
$$- \frac{4}{225} + \left(- \frac{59}{2278125 \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}} + \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}\right) \log{\left(- \frac{29615625 \left(- \frac{59}{2278125 \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}} + \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}\right)^{2}}{7379} - \frac{11918}{4980825 \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}} + \frac{5845}{7379} + \frac{681750 \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}}{7379} \right)} - \left(- \frac{59}{2278125 \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}} + \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}\right) \log{\left(- \frac{29615625 \left(- \frac{59}{2278125 \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}} + \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}\right)^{2}}{7379} - \frac{1534}{7379} - \frac{11918}{4980825 \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}} + \frac{681750 \sqrt[3]{\frac{131}{123018750} + \frac{7379 \sqrt{5}}{15377343750}}}{7379} \right)}$$ 
-4/225 + ((131/123018750 + 7379*sqrt(5)/15377343750)^(1/3) - 59/(2278125*(131/123018750 + 7379*sqrt(5)/15377343750)^(1/3)))*log(5845/7379 - 29615625*((131/123018750 + 7379*sqrt(5)/15377343750)^(1/3) - 59/(2278125*(131/123018750 + 7379*sqrt(5)/15377343750)^(1/3)))^2/7379 - 11918/(4980825*(131/123018750 + 7379*sqrt(5)/15377343750)^(1/3)) + 681750*(131/123018750 + 7379*sqrt(5)/15377343750)^(1/3)/7379) - ((131/123018750 + 7379*sqrt(5)/15377343750)^(1/3) - 59/(2278125*(131/123018750 + 7379*sqrt(5)/15377343750)^(1/3)))*log(-1534/7379 - 29615625*((131/123018750 + 7379*sqrt(5)/15377343750)^(1/3) - 59/(2278125*(131/123018750 + 7379*sqrt(5)/15377343750)^(1/3)))^2/7379 - 11918/(4980825*(131/123018750 + 7379*sqrt(5)/15377343750)^(1/3)) + 681750*(131/123018750 + 7379*sqrt(5)/15377343750)^(1/3)/7379)
Respuesta numérica [src] 
0.0103662359530387
0.0103662359530387

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

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