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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  2                    
  /                    
 |                     
 |         1           
 |  ---------------- dx
 |    / 2          \   
 |  x*\x  - 4*x + 1/   
 |                     
/                      
1
∫121x((x2−4x)+1) dx\int\limits_{1}^{2} \frac{1}{x \left(\left(x^{2} - 4 x\right) + 1\right)}\, dx 
Integral(1/(x*(x^2 - 4*x + 1)), (x, 1, 2))
Respuesta (Indefinida) [src] 
                               //            /  ___         \                    \                             
                               ||   ___      |\/ 3 *(-2 + x)|                    |                             
                               ||-\/ 3 *acoth|--------------|                    |                             
  /                            ||            \      3       /               2    |                             
 |                             ||-----------------------------  for (-2 + x)  > 3|      /     2      \         
 |        1                    ||              3                                 |   log\1 + x  - 4*x/         
 | ---------------- dx = C + 2*|<                                                | - ----------------- + log(x)
 |   / 2          \            ||            /  ___         \                    |           2                 
 | x*\x  - 4*x + 1/            ||   ___      |\/ 3 *(-2 + x)|                    |                             
 |                             ||-\/ 3 *atanh|--------------|                    |                             
/                              ||            \      3       /               2    |                             
                               ||-----------------------------  for (-2 + x)  < 3|                             
                               \\              3                                 /
∫1x((x2−4x)+1) dx=C+2({−3acoth⁡(3(x−2)3)3for (x−2)2>3−3atanh⁡(3(x−2)3)3for (x−2)2<3)+log⁡(x)−log⁡(x2−4x+1)2\int \frac{1}{x \left(\left(x^{2} - 4 x\right) + 1\right)}\, dx = C + 2 \left(\begin{cases} - \frac{\sqrt{3} \operatorname{acoth}{\left(\frac{\sqrt{3} \left(x - 2\right)}{3} \right)}}{3} & \text{for}\: \left(x - 2\right)^{2} > 3 \\- \frac{\sqrt{3} \operatorname{atanh}{\left(\frac{\sqrt{3} \left(x - 2\right)}{3} \right)}}{3} & \text{for}\: \left(x - 2\right)^{2} < 3 \end{cases}\right) + \log{\left(x \right)} - \frac{\log{\left(x^{2} - 4 x + 1 \right)}}{2}
Simplificar
Gráfica
1.002.001.101.201.301.401.501.601.701.801.90-1.00.0
Trazar el gráfico f(x)
Respuesta [src] 
                 /                                 2\                 /          /                     2          \\                    /                                  2\                 /          /                      2          \\         
                 |                    /        ___\ |                 |          |        /        ___\           ||                    |                     /        ___\ |                 |          |         /        ___\           ||         
                 |                    |  1   \/ 3 | |                 |          |        |  1   \/ 3 |           ||                    |                     |  1   \/ 3 | |                 |          |         |  1   \/ 3 |           ||         
/        ___\    |           ___   78*|- - - -----| |   /        ___\ |          |     78*|- - + -----|        ___||   /        ___\    |            ___   78*|- - - -----| |   /        ___\ |          |      78*|- - + -----|        ___||         
|  1   \/ 3 |    |  91   3*\/ 3       \  2     3  / |   |  1   \/ 3 | |          |91      \  2     3  /    3*\/ 3 ||   |  1   \/ 3 |    |  137   3*\/ 3       \  2     3  / |   |  1   \/ 3 | |          |137      \  2     3  /    3*\/ 3 ||         
|- - - -----|*log|- -- - ------- + -----------------| + |- - + -----|*|pi*I + log|-- - ----------------- - -------|| - |- - - -----|*log|- --- - ------- + -----------------| - |- - + -----|*|pi*I + log|--- - ----------------- - -------|| + log(2)
\  2     3  /    \  46      23             23       /   \  2     3  / \          \46           23             23  //   \  2     3  /    \   46      23             23       /   \  2     3  / \          \ 46           23             23  //
(−33−12)log⁡(−9146−3323+78(−33−12)223)−(−33−12)log⁡(−13746−3323+78(−33−12)223)+log⁡(2)−(−12+33)(log⁡(−3323−78(−12+33)223+13746)+iπ)+(−12+33)(log⁡(−3323−78(−12+33)223+9146)+iπ)\left(- \frac{\sqrt{3}}{3} - \frac{1}{2}\right) \log{\left(- \frac{91}{46} - \frac{3 \sqrt{3}}{23} + \frac{78 \left(- \frac{\sqrt{3}}{3} - \frac{1}{2}\right)^{2}}{23} \right)} - \left(- \frac{\sqrt{3}}{3} - \frac{1}{2}\right) \log{\left(- \frac{137}{46} - \frac{3 \sqrt{3}}{23} + \frac{78 \left(- \frac{\sqrt{3}}{3} - \frac{1}{2}\right)^{2}}{23} \right)} + \log{\left(2 \right)} - \left(- \frac{1}{2} + \frac{\sqrt{3}}{3}\right) \left(\log{\left(- \frac{3 \sqrt{3}}{23} - \frac{78 \left(- \frac{1}{2} + \frac{\sqrt{3}}{3}\right)^{2}}{23} + \frac{137}{46} \right)} + i \pi\right) + \left(- \frac{1}{2} + \frac{\sqrt{3}}{3}\right) \left(\log{\left(- \frac{3 \sqrt{3}}{23} - \frac{78 \left(- \frac{1}{2} + \frac{\sqrt{3}}{3}\right)^{2}}{23} + \frac{91}{46} \right)} + i \pi\right) 
=
= 
                 /                                 2\                 /          /                     2          \\                    /                                  2\                 /          /                      2          \\         
                 |                    /        ___\ |                 |          |        /        ___\           ||                    |                     /        ___\ |                 |          |         /        ___\           ||         
                 |                    |  1   \/ 3 | |                 |          |        |  1   \/ 3 |           ||                    |                     |  1   \/ 3 | |                 |          |         |  1   \/ 3 |           ||         
/        ___\    |           ___   78*|- - - -----| |   /        ___\ |          |     78*|- - + -----|        ___||   /        ___\    |            ___   78*|- - - -----| |   /        ___\ |          |      78*|- - + -----|        ___||         
|  1   \/ 3 |    |  91   3*\/ 3       \  2     3  / |   |  1   \/ 3 | |          |91      \  2     3  /    3*\/ 3 ||   |  1   \/ 3 |    |  137   3*\/ 3       \  2     3  / |   |  1   \/ 3 | |          |137      \  2     3  /    3*\/ 3 ||         
|- - - -----|*log|- -- - ------- + -----------------| + |- - + -----|*|pi*I + log|-- - ----------------- - -------|| - |- - - -----|*log|- --- - ------- + -----------------| - |- - + -----|*|pi*I + log|--- - ----------------- - -------|| + log(2)
\  2     3  /    \  46      23             23       /   \  2     3  / \          \46           23             23  //   \  2     3  /    \   46      23             23       /   \  2     3  / \          \ 46           23             23  //
(−33−12)log⁡(−9146−3323+78(−33−12)223)−(−33−12)log⁡(−13746−3323+78(−33−12)223)+log⁡(2)−(−12+33)(log⁡(−3323−78(−12+33)223+13746)+iπ)+(−12+33)(log⁡(−3323−78(−12+33)223+9146)+iπ)\left(- \frac{\sqrt{3}}{3} - \frac{1}{2}\right) \log{\left(- \frac{91}{46} - \frac{3 \sqrt{3}}{23} + \frac{78 \left(- \frac{\sqrt{3}}{3} - \frac{1}{2}\right)^{2}}{23} \right)} - \left(- \frac{\sqrt{3}}{3} - \frac{1}{2}\right) \log{\left(- \frac{137}{46} - \frac{3 \sqrt{3}}{23} + \frac{78 \left(- \frac{\sqrt{3}}{3} - \frac{1}{2}\right)^{2}}{23} \right)} + \log{\left(2 \right)} - \left(- \frac{1}{2} + \frac{\sqrt{3}}{3}\right) \left(\log{\left(- \frac{3 \sqrt{3}}{23} - \frac{78 \left(- \frac{1}{2} + \frac{\sqrt{3}}{3}\right)^{2}}{23} + \frac{137}{46} \right)} + i \pi\right) + \left(- \frac{1}{2} + \frac{\sqrt{3}}{3}\right) \left(\log{\left(- \frac{3 \sqrt{3}}{23} - \frac{78 \left(- \frac{1}{2} + \frac{\sqrt{3}}{3}\right)^{2}}{23} + \frac{91}{46} \right)} + i \pi\right) 
(-1/2 - sqrt(3)/3)*log(-91/46 - 3*sqrt(3)/23 + 78*(-1/2 - sqrt(3)/3)^2/23) + (-1/2 + sqrt(3)/3)*(pi*i + log(91/46 - 78*(-1/2 + sqrt(3)/3)^2/23 - 3*sqrt(3)/23)) - (-1/2 - sqrt(3)/3)*log(-137/46 - 3*sqrt(3)/23 + 78*(-1/2 - sqrt(3)/3)^2/23) - (-1/2 + sqrt(3)/3)*(pi*i + log(137/46 - 78*(-1/2 + sqrt(3)/3)^2/23 - 3*sqrt(3)/23)) + log(2)
Respuesta numérica [src] 
-0.269931369795083
-0.269931369795083

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

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