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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                
  /                
 |                 
 |       x         
 |  ------------ dx
 |   2             
 |  x  + 4*x + 1   
 |                 
/                  
0
∫01x(x2+4x)+1 dx\int\limits_{0}^{1} \frac{x}{\left(x^{2} + 4 x\right) + 1}\, dx 
Integral(x/(x^2 + 4*x + 1), (x, 0, 1))
Respuesta (Indefinida) [src] 
                                               //            /  ___        \                   \
                                               ||   ___      |\/ 3 *(2 + x)|                   |
                                               ||-\/ 3 *acoth|-------------|                   |
  /                                            ||            \      3      /              2    |
 |                          /     2      \     ||----------------------------  for (2 + x)  > 3|
 |      x                log\1 + x  + 4*x/     ||             3                                |
 | ------------ dx = C + ----------------- - 2*|<                                              |
 |  2                            2             ||            /  ___        \                   |
 | x  + 4*x + 1                                ||   ___      |\/ 3 *(2 + x)|                   |
 |                                             ||-\/ 3 *atanh|-------------|                   |
/                                              ||            \      3      /              2    |
                                               ||----------------------------  for (2 + x)  < 3|
                                               \\             3                                /
∫x(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⁡(x2+4x+1)2\int \frac{x}{\left(x^{2} + 4 x\right) + 1}\, 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) + \frac{\log{\left(x^{2} + 4 x + 1 \right)}}{2}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.9002
Trazar el gráfico f(x)
Respuesta [src] 
/      ___\                  /      ___\                  /      ___\                  /      ___\               
|1   \/ 3 |    /      ___\   |1   \/ 3 |    /      ___\   |1   \/ 3 |    /      ___\   |1   \/ 3 |    /      ___\
|- - -----|*log\3 - \/ 3 / + |- + -----|*log\3 + \/ 3 / - |- - -----|*log\2 - \/ 3 / - |- + -----|*log\2 + \/ 3 /
\2     3  /                  \2     3  /                  \2     3  /                  \2     3  /
−(12+33)log⁡(3+2)−(12−33)log⁡(2−3)+(12−33)log⁡(3−3)+(12+33)log⁡(3+3)- \left(\frac{1}{2} + \frac{\sqrt{3}}{3}\right) \log{\left(\sqrt{3} + 2 \right)} - \left(\frac{1}{2} - \frac{\sqrt{3}}{3}\right) \log{\left(2 - \sqrt{3} \right)} + \left(\frac{1}{2} - \frac{\sqrt{3}}{3}\right) \log{\left(3 - \sqrt{3} \right)} + \left(\frac{1}{2} + \frac{\sqrt{3}}{3}\right) \log{\left(\sqrt{3} + 3 \right)} 
=
= 
/      ___\                  /      ___\                  /      ___\                  /      ___\               
|1   \/ 3 |    /      ___\   |1   \/ 3 |    /      ___\   |1   \/ 3 |    /      ___\   |1   \/ 3 |    /      ___\
|- - -----|*log\3 - \/ 3 / + |- + -----|*log\3 + \/ 3 / - |- - -----|*log\2 - \/ 3 / - |- + -----|*log\2 + \/ 3 /
\2     3  /                  \2     3  /                  \2     3  /                  \2     3  /
−(12+33)log⁡(3+2)−(12−33)log⁡(2−3)+(12−33)log⁡(3−3)+(12+33)log⁡(3+3)- \left(\frac{1}{2} + \frac{\sqrt{3}}{3}\right) \log{\left(\sqrt{3} + 2 \right)} - \left(\frac{1}{2} - \frac{\sqrt{3}}{3}\right) \log{\left(2 - \sqrt{3} \right)} + \left(\frac{1}{2} - \frac{\sqrt{3}}{3}\right) \log{\left(3 - \sqrt{3} \right)} + \left(\frac{1}{2} + \frac{\sqrt{3}}{3}\right) \log{\left(\sqrt{3} + 3 \right)} 
(1/2 - sqrt(3)/3)*log(3 - sqrt(3)) + (1/2 + sqrt(3)/3)*log(3 + sqrt(3)) - (1/2 - sqrt(3)/3)*log(2 - sqrt(3)) - (1/2 + sqrt(3)/3)*log(2 + sqrt(3))
Respuesta numérica [src] 
0.135533738313081
0.135533738313081

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

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