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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                            
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 |             1               
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 |            2                
 |  / 2      \             2   
 |  \x  + 3*x/  + 2*x + 3*x    
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0
∫011(x2+3x)2+(3x2+2x) dx\int\limits_{0}^{1} \frac{1}{\left(x^{2} + 3 x\right)^{2} + \left(3 x^{2} + 2 x\right)}\, dx 
Integral(1/((x^2 + 3*x)^2 + 2*x + 3*x^2), (x, 0, 1))
Respuesta (Indefinida) [src] 
                                                                                                                                          /                                  2                   \                                                                                                                                                                                                                                                                                                    /                                                      ____________                                                                                                           ____________                                                                                                        ____________                                               \
                                                                                                                                          |            /       2/3     3 ___\                    |                                                                                                                                                                                                                                                                        _______________________     |                                                     /      3 ___                                                                                                   3 ___   /      3 ___                                                                                                        /      3 ___                                                |
                                                                                                                                          |            |1   2*6      2*\/ 6 |                    |   /    3 ___    2/3\    /                                                       2/3                    3 ___       \                                                                                                                                            ___   /         2/3     3 ___      |                                                28*\/  -1 + \/ 6                                                                                                  7*\/ 6 *\/  -1 + \/ 6                                                                                                  14*x*\/  -1 + \/ 6                                                 |
                                                                                                                /       2/3     3 ___\    |         81*|- - ------ + -------|       2/3     3 ___|   |1   \/ 6    6   |    | 2             2               /    3 ___\            6                     4*\/ 6        |                                                                                                  /  ___      2/3 6 ___      2/3 6 ___\   \/ 3 *\/  -6 + 5*6    + 6*\/ 6  *atan|------------------------------------------------------------------------------------------------------------------ + ------------------------------------------------------------------------------------------------------------------ + ------------------------------------------------------------------------------------------------------------------|
  /                                                                                                             |1   2*6      2*\/ 6 |    |17          \3     9         9   /    2*6      2*\/ 6 |   |- - ----- + ----|*log|x  - --------------------- + x*\4 + \/ 6 / + --------------------- - ---------------------|                                                                                   3 ___  5/6     |\/ 3    2*2   *\/ 3    x*2   *\/ 3 |                                        |         _______________________                 _______________________                   _______________________            _______________________                 _______________________                   _______________________            _______________________                 _______________________                   _______________________|
 |                                                     /                             /             2       \\   |- - ------ + -------|*log|-- + x - -------------------------- - ------ + -------|   \3     9      9  /    |            3 ___      2/3                          3 ___      2/3          3 ___      2/3|   3 ___    /        3 ___\   3 ___    /     2/3    2     3 ___     /    3 ___\\   \/ 2 *3   *atan|----- + ------------ + ------------|                                        |  ___   /         2/3     3 ___    3 ___  5/6   /         2/3     3 ___       2/3 6 ___   /         2/3     3 ___      ___   /         2/3     3 ___    3 ___  5/6   /         2/3     3 ___       2/3 6 ___   /         2/3     3 ___      ___   /         2/3     3 ___    3 ___  5/6   /         2/3     3 ___       2/3 6 ___   /         2/3     3 ___ |
 |            1                      log(x)            |     3                       |2       162*t    36*t||   \3     9         9   /    \28                   28                 7         7   /                         \     -5 - 3*\/ 6  + 3*6                      -5 - 3*\/ 6  + 3*6      -5 - 3*\/ 6  + 3*6   /   \/ 6 *log\2 + x - \/ 6 /   \/ 6 *log\4 + 6    + x  + 2*\/ 6  + x*\4 + \/ 6 //                  \  3          3              3      /                                        \\/ 3 *\/  -6 + 5*6    + 6*\/ 6   - \/ 2 *3   *\/  -6 + 5*6    + 6*\/ 6   + 3*2   *\/ 3 *\/  -6 + 5*6    + 6*\/ 6     \/ 3 *\/  -6 + 5*6    + 6*\/ 6   - \/ 2 *3   *\/  -6 + 5*6    + 6*\/ 6   + 3*2   *\/ 3 *\/  -6 + 5*6    + 6*\/ 6     \/ 3 *\/  -6 + 5*6    + 6*\/ 6   - \/ 2 *3   *\/  -6 + 5*6    + 6*\/ 6   + 3*2   *\/ 3 *\/  -6 + 5*6    + 6*\/ 6  /
 | ------------------------ dx = C + ------ - 3*RootSum|486*t  + 54*t + 1, t -> t*log|- + x - ------ + ----|| - ---------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------ - ------------------------ + -------------------------------------------------- + ---------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
 |           2                         2               \                             \7         7       7  //                                           2                                                                                                    2                                                                       3                                       6                                                     3                                                                                                                                                                                                                           ____________                                                                                                                                                                                         
 | / 2      \             2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   /      3 ___                                                                                                                                                                                          
 | \x  + 3*x/  + 2*x + 3*x                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                9*\/  -1 + \/ 6                                                                                                                                                                                           
 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  
/
∫1(x2+3x)2+(3x2+2x) dx=C+log⁡(x)2−63log⁡(x−63+2)3−(−2⋅6239+13+2639)log⁡(x−2⋅6237−81(−2⋅6239+13+2639)228+2637+1728)2+63log⁡(x2+x(63+4)+623+263+4)6−(−639+13+6239)log⁡(x2+x(63+4)+623−363−5+3⋅623−2−363−5+3⋅623−463−363−5+3⋅623)2+23⋅356atan⁡(22336x3+33+2⋅223363)3+3−6+663+5⋅623atan⁡(14x−1+63−23⋅356−6+663+5⋅623+3−6+663+5⋅623+3⋅22336−6+663+5⋅623+763−1+63−23⋅356−6+663+5⋅623+3−6+663+5⋅623+3⋅22336−6+663+5⋅623+28−1+63−23⋅356−6+663+5⋅623+3−6+663+5⋅623+3⋅22336−6+663+5⋅623)9−1+63−3RootSum⁡(486t3+54t+1,(t↦tlog⁡(−162t27+36t7+x+27)))\int \frac{1}{\left(x^{2} + 3 x\right)^{2} + \left(3 x^{2} + 2 x\right)}\, dx = C + \frac{\log{\left(x \right)}}{2} - \frac{\sqrt[3]{6} \log{\left(x - \sqrt[3]{6} + 2 \right)}}{3} - \frac{\left(- \frac{2 \cdot 6^{\frac{2}{3}}}{9} + \frac{1}{3} + \frac{2 \sqrt[3]{6}}{9}\right) \log{\left(x - \frac{2 \cdot 6^{\frac{2}{3}}}{7} - \frac{81 \left(- \frac{2 \cdot 6^{\frac{2}{3}}}{9} + \frac{1}{3} + \frac{2 \sqrt[3]{6}}{9}\right)^{2}}{28} + \frac{2 \sqrt[3]{6}}{7} + \frac{17}{28} \right)}}{2} + \frac{\sqrt[3]{6} \log{\left(x^{2} + x \left(\sqrt[3]{6} + 4\right) + 6^{\frac{2}{3}} + 2 \sqrt[3]{6} + 4 \right)}}{6} - \frac{\left(- \frac{\sqrt[3]{6}}{9} + \frac{1}{3} + \frac{6^{\frac{2}{3}}}{9}\right) \log{\left(x^{2} + x \left(\sqrt[3]{6} + 4\right) + \frac{6^{\frac{2}{3}}}{- 3 \sqrt[3]{6} - 5 + 3 \cdot 6^{\frac{2}{3}}} - \frac{2}{- 3 \sqrt[3]{6} - 5 + 3 \cdot 6^{\frac{2}{3}}} - \frac{4 \sqrt[3]{6}}{- 3 \sqrt[3]{6} - 5 + 3 \cdot 6^{\frac{2}{3}}} \right)}}{2} + \frac{\sqrt[3]{2} \cdot 3^{\frac{5}{6}} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt[6]{3} x}{3} + \frac{\sqrt{3}}{3} + \frac{2 \cdot 2^{\frac{2}{3}} \sqrt[6]{3}}{3} \right)}}{3} + \frac{\sqrt{3} \sqrt{-6 + 6 \sqrt[3]{6} + 5 \cdot 6^{\frac{2}{3}}} \operatorname{atan}{\left(\frac{14 x \sqrt{-1 + \sqrt[3]{6}}}{- \sqrt[3]{2} \cdot 3^{\frac{5}{6}} \sqrt{-6 + 6 \sqrt[3]{6} + 5 \cdot 6^{\frac{2}{3}}} + \sqrt{3} \sqrt{-6 + 6 \sqrt[3]{6} + 5 \cdot 6^{\frac{2}{3}}} + 3 \cdot 2^{\frac{2}{3}} \sqrt[6]{3} \sqrt{-6 + 6 \sqrt[3]{6} + 5 \cdot 6^{\frac{2}{3}}}} + \frac{7 \sqrt[3]{6} \sqrt{-1 + \sqrt[3]{6}}}{- \sqrt[3]{2} \cdot 3^{\frac{5}{6}} \sqrt{-6 + 6 \sqrt[3]{6} + 5 \cdot 6^{\frac{2}{3}}} + \sqrt{3} \sqrt{-6 + 6 \sqrt[3]{6} + 5 \cdot 6^{\frac{2}{3}}} + 3 \cdot 2^{\frac{2}{3}} \sqrt[6]{3} \sqrt{-6 + 6 \sqrt[3]{6} + 5 \cdot 6^{\frac{2}{3}}}} + \frac{28 \sqrt{-1 + \sqrt[3]{6}}}{- \sqrt[3]{2} \cdot 3^{\frac{5}{6}} \sqrt{-6 + 6 \sqrt[3]{6} + 5 \cdot 6^{\frac{2}{3}}} + \sqrt{3} \sqrt{-6 + 6 \sqrt[3]{6} + 5 \cdot 6^{\frac{2}{3}}} + 3 \cdot 2^{\frac{2}{3}} \sqrt[6]{3} \sqrt{-6 + 6 \sqrt[3]{6} + 5 \cdot 6^{\frac{2}{3}}}} \right)}}{9 \sqrt{-1 + \sqrt[3]{6}}} - 3 \operatorname{RootSum} {\left(486 t^{3} + 54 t + 1, \left( t \mapsto t \log{\left(- \frac{162 t^{2}}{7} + \frac{36 t}{7} + x + \frac{2}{7} \right)} \right)\right)}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.90-50005000
Trazar el gráfico f(x)
Respuesta [src] 
  1                            
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 |             1               
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 |            2                
 |  / 2      \             2   
 |  \x  + 3*x/  + 2*x + 3*x    
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/                              
0
∫0113x2+2x+(x2+3x)2 dx\int\limits_{0}^{1} \frac{1}{3 x^{2} + 2 x + \left(x^{2} + 3 x\right)^{2}}\, dx 
=
= 
  1                            
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 |             1               
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 |            2                
 |  / 2      \             2   
 |  \x  + 3*x/  + 2*x + 3*x    
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/                              
0
∫0113x2+2x+(x2+3x)2 dx\int\limits_{0}^{1} \frac{1}{3 x^{2} + 2 x + \left(x^{2} + 3 x\right)^{2}}\, dx 
Integral(1/((x^2 + 3*x)^2 + 2*x + 3*x^2), (x, 0, 1))
Respuesta numérica [src] 
21.0309061535901
21.0309061535901

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

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