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Resolución de integrales/ 1-2*x/ x^(2)/ (1-2*x)/(x^(4)-2x^(2)-3)

Integral de (1-2*x)/(x^(4)-2x^(2)-3) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                 
  /                 
 |                  
 |     1 - 2*x      
 |  ------------- dx
 |   4      2       
 |  x  - 2*x  - 3   
 |                  
/                   
0
∫011−2x(x4−2x2)−3 dx\int\limits_{0}^{1} \frac{1 - 2 x}{\left(x^{4} - 2 x^{2}\right) - 3}\, dx 
Integral((1 - 2*x)/(x^4 - 2*x^2 - 3), (x, 0, 1))
Respuesta (Indefinida) [src] 
                                                   /            /    ___\                           
                                                   |   ___      |x*\/ 3 |                           
                                                   |-\/ 3 *acoth|-------|                           
                                                   |            \   3   /        2                  
                                                   |----------------------  for x  > 3              
                                                   |          3                                     
                                                   <                                                
                                                   |            /    ___\                           
                                                   |   ___      |x*\/ 3 |                           
                                                   |-\/ 3 *atanh|-------|                           
  /                                                |            \   3   /        2                  
 |                                     /      2\   |----------------------  for x  < 3      /     2\
 |    1 - 2*x             atan(x)   log\-3 + x /   \          3                          log\1 + x /
 | ------------- dx = C - ------- - ------------ + ----------------------------------- + -----------
 |  4      2                 4           4                          4                         4     
 | x  - 2*x  - 3                                                                                    
 |                                                                                                  
/
∫1−2x(x4−2x2)−3 dx=C+{−3acoth⁡(3x3)3for x2>3−3atanh⁡(3x3)3for x2<34−log⁡(x2−3)4+log⁡(x2+1)4−atan⁡(x)4\int \frac{1 - 2 x}{\left(x^{4} - 2 x^{2}\right) - 3}\, dx = C + \frac{\begin{cases} - \frac{\sqrt{3} \operatorname{acoth}{\left(\frac{\sqrt{3} x}{3} \right)}}{3} & \text{for}\: x^{2} > 3 \\- \frac{\sqrt{3} \operatorname{atanh}{\left(\frac{\sqrt{3} x}{3} \right)}}{3} & \text{for}\: x^{2} < 3 \end{cases}}{4} - \frac{\log{\left(x^{2} - 3 \right)}}{4} + \frac{\log{\left(x^{2} + 1 \right)}}{4} - \frac{\operatorname{atan}{\left(x \right)}}{4}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.900.5-0.5
Trazar el gráfico f(x)
Respuesta [src] 
                            /          /                       3                   2            \\                  /                                    2                    3\               /          /                       3                   2            \\                  /                                    2                    3\
                            |          |            /      ___\         /      ___\             ||                  |                         /      ___\          /      ___\ |               |          |            /      ___\         /      ___\             ||                  |                         /      ___\          /      ___\ |
                            |          |            |1   \/ 3 |         |1   \/ 3 |             ||                  |                         |1   \/ 3 |          |1   \/ 3 | |               |          |            |1   \/ 3 |         |1   \/ 3 |             ||                  |                         |1   \/ 3 |          |1   \/ 3 | |
                /      ___\ |          |      19200*|- - -----|    2112*|- - -----|          ___||   /      ___\    |              ___   2112*|- + -----|    19200*|- + -----| |   /      ___\ |          |      19200*|- - -----|    2112*|- - -----|          ___||   /      ___\    |              ___   2112*|- + -----|    19200*|- + -----| |
  pi   log(2)   |1   \/ 3 | |          |518         \4     24 /         \4     24 /    247*\/ 3 ||   |1   \/ 3 |    |  518   247*\/ 3         \4     24 /          \4     24 / |   |1   \/ 3 | |          |361         \4     24 /         \4     24 /    247*\/ 3 ||   |1   \/ 3 |    |  361   247*\/ 3         \4     24 /          \4     24 / |
- -- + ------ + |- - -----|*|pi*I + log|--- - ------------------ - ----------------- - ---------|| + |- + -----|*log|- --- - --------- + ----------------- + ------------------| - |- - -----|*|pi*I + log|--- - ------------------ - ----------------- - ---------|| - |- + -----|*log|- --- - --------- + ----------------- + ------------------|
  16     4      \4     24 / \          \157          157                  157             942   //   \4     24 /    \  157      942             157                 157        /   \4     24 / \          \157          157                  157             942   //   \4     24 /    \  157      942             157                 157        /
−(324+14)log⁡(−361157−2473942+2112(324+14)2157+19200(324+14)3157)−π16+log⁡(2)4+(324+14)log⁡(−518157−2473942+2112(324+14)2157+19200(324+14)3157)−(14−324)(log⁡(−19200(14−324)3157−2473942−2112(14−324)2157+361157)+iπ)+(14−324)(log⁡(−19200(14−324)3157−2473942−2112(14−324)2157+518157)+iπ)- \left(\frac{\sqrt{3}}{24} + \frac{1}{4}\right) \log{\left(- \frac{361}{157} - \frac{247 \sqrt{3}}{942} + \frac{2112 \left(\frac{\sqrt{3}}{24} + \frac{1}{4}\right)^{2}}{157} + \frac{19200 \left(\frac{\sqrt{3}}{24} + \frac{1}{4}\right)^{3}}{157} \right)} - \frac{\pi}{16} + \frac{\log{\left(2 \right)}}{4} + \left(\frac{\sqrt{3}}{24} + \frac{1}{4}\right) \log{\left(- \frac{518}{157} - \frac{247 \sqrt{3}}{942} + \frac{2112 \left(\frac{\sqrt{3}}{24} + \frac{1}{4}\right)^{2}}{157} + \frac{19200 \left(\frac{\sqrt{3}}{24} + \frac{1}{4}\right)^{3}}{157} \right)} - \left(\frac{1}{4} - \frac{\sqrt{3}}{24}\right) \left(\log{\left(- \frac{19200 \left(\frac{1}{4} - \frac{\sqrt{3}}{24}\right)^{3}}{157} - \frac{247 \sqrt{3}}{942} - \frac{2112 \left(\frac{1}{4} - \frac{\sqrt{3}}{24}\right)^{2}}{157} + \frac{361}{157} \right)} + i \pi\right) + \left(\frac{1}{4} - \frac{\sqrt{3}}{24}\right) \left(\log{\left(- \frac{19200 \left(\frac{1}{4} - \frac{\sqrt{3}}{24}\right)^{3}}{157} - \frac{247 \sqrt{3}}{942} - \frac{2112 \left(\frac{1}{4} - \frac{\sqrt{3}}{24}\right)^{2}}{157} + \frac{518}{157} \right)} + i \pi\right) 
=
= 
                            /          /                       3                   2            \\                  /                                    2                    3\               /          /                       3                   2            \\                  /                                    2                    3\
                            |          |            /      ___\         /      ___\             ||                  |                         /      ___\          /      ___\ |               |          |            /      ___\         /      ___\             ||                  |                         /      ___\          /      ___\ |
                            |          |            |1   \/ 3 |         |1   \/ 3 |             ||                  |                         |1   \/ 3 |          |1   \/ 3 | |               |          |            |1   \/ 3 |         |1   \/ 3 |             ||                  |                         |1   \/ 3 |          |1   \/ 3 | |
                /      ___\ |          |      19200*|- - -----|    2112*|- - -----|          ___||   /      ___\    |              ___   2112*|- + -----|    19200*|- + -----| |   /      ___\ |          |      19200*|- - -----|    2112*|- - -----|          ___||   /      ___\    |              ___   2112*|- + -----|    19200*|- + -----| |
  pi   log(2)   |1   \/ 3 | |          |518         \4     24 /         \4     24 /    247*\/ 3 ||   |1   \/ 3 |    |  518   247*\/ 3         \4     24 /          \4     24 / |   |1   \/ 3 | |          |361         \4     24 /         \4     24 /    247*\/ 3 ||   |1   \/ 3 |    |  361   247*\/ 3         \4     24 /          \4     24 / |
- -- + ------ + |- - -----|*|pi*I + log|--- - ------------------ - ----------------- - ---------|| + |- + -----|*log|- --- - --------- + ----------------- + ------------------| - |- - -----|*|pi*I + log|--- - ------------------ - ----------------- - ---------|| - |- + -----|*log|- --- - --------- + ----------------- + ------------------|
  16     4      \4     24 / \          \157          157                  157             942   //   \4     24 /    \  157      942             157                 157        /   \4     24 / \          \157          157                  157             942   //   \4     24 /    \  157      942             157                 157        /
−(324+14)log⁡(−361157−2473942+2112(324+14)2157+19200(324+14)3157)−π16+log⁡(2)4+(324+14)log⁡(−518157−2473942+2112(324+14)2157+19200(324+14)3157)−(14−324)(log⁡(−19200(14−324)3157−2473942−2112(14−324)2157+361157)+iπ)+(14−324)(log⁡(−19200(14−324)3157−2473942−2112(14−324)2157+518157)+iπ)- \left(\frac{\sqrt{3}}{24} + \frac{1}{4}\right) \log{\left(- \frac{361}{157} - \frac{247 \sqrt{3}}{942} + \frac{2112 \left(\frac{\sqrt{3}}{24} + \frac{1}{4}\right)^{2}}{157} + \frac{19200 \left(\frac{\sqrt{3}}{24} + \frac{1}{4}\right)^{3}}{157} \right)} - \frac{\pi}{16} + \frac{\log{\left(2 \right)}}{4} + \left(\frac{\sqrt{3}}{24} + \frac{1}{4}\right) \log{\left(- \frac{518}{157} - \frac{247 \sqrt{3}}{942} + \frac{2112 \left(\frac{\sqrt{3}}{24} + \frac{1}{4}\right)^{2}}{157} + \frac{19200 \left(\frac{\sqrt{3}}{24} + \frac{1}{4}\right)^{3}}{157} \right)} - \left(\frac{1}{4} - \frac{\sqrt{3}}{24}\right) \left(\log{\left(- \frac{19200 \left(\frac{1}{4} - \frac{\sqrt{3}}{24}\right)^{3}}{157} - \frac{247 \sqrt{3}}{942} - \frac{2112 \left(\frac{1}{4} - \frac{\sqrt{3}}{24}\right)^{2}}{157} + \frac{361}{157} \right)} + i \pi\right) + \left(\frac{1}{4} - \frac{\sqrt{3}}{24}\right) \left(\log{\left(- \frac{19200 \left(\frac{1}{4} - \frac{\sqrt{3}}{24}\right)^{3}}{157} - \frac{247 \sqrt{3}}{942} - \frac{2112 \left(\frac{1}{4} - \frac{\sqrt{3}}{24}\right)^{2}}{157} + \frac{518}{157} \right)} + i \pi\right) 
-pi/16 + log(2)/4 + (1/4 - sqrt(3)/24)*(pi*i + log(518/157 - 19200*(1/4 - sqrt(3)/24)^3/157 - 2112*(1/4 - sqrt(3)/24)^2/157 - 247*sqrt(3)/942)) + (1/4 + sqrt(3)/24)*log(-518/157 - 247*sqrt(3)/942 + 2112*(1/4 + sqrt(3)/24)^2/157 + 19200*(1/4 + sqrt(3)/24)^3/157) - (1/4 - sqrt(3)/24)*(pi*i + log(361/157 - 19200*(1/4 - sqrt(3)/24)^3/157 - 2112*(1/4 - sqrt(3)/24)^2/157 - 247*sqrt(3)/942)) - (1/4 + sqrt(3)/24)*log(-361/157 - 247*sqrt(3)/942 + 2112*(1/4 + sqrt(3)/24)^2/157 + 19200*(1/4 + sqrt(3)/24)^3/157)
Respuesta numérica [src] 
-0.0167397182199529
-0.0167397182199529

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

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