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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                
  /                
 |                 
 |    3*x + 1      
 |  ------------ dx
 |   2             
 |  x  + 4*x + 1   
 |                 
/                  
0
∫013x+1(x2+4x)+1 dx\int\limits_{0}^{1} \frac{3 x + 1}{\left(x^{2} + 4 x\right) + 1}\, dx 
Integral((3*x + 1)/(x^2 + 4*x + 1), (x, 0, 1))
Respuesta (Indefinida) [src] 
                           //            /  ___        \                   \                      
                           ||   ___      |\/ 3 *(2 + x)|                   |                      
                           ||-\/ 3 *acoth|-------------|                   |                      
  /                        ||            \      3      /              2    |                      
 |                         ||----------------------------  for (2 + x)  > 3|        /     2      \
 |   3*x + 1               ||             3                                |   3*log\1 + x  + 4*x/
 | ------------ dx = C - 5*|<                                              | + -------------------
 |  2                      ||            /  ___        \                   |            2         
 | x  + 4*x + 1            ||   ___      |\/ 3 *(2 + x)|                   |                      
 |                         ||-\/ 3 *atanh|-------------|                   |                      
/                          ||            \      3      /              2    |                      
                           ||----------------------------  for (2 + x)  < 3|                      
                           \\             3                                /
∫3x+1(x2+4x)+1 dx=C−5({−3acoth⁡(3(x+2)3)3for (x+2)2>3−3atanh⁡(3(x+2)3)3for (x+2)2<3)+3log⁡(x2+4x+1)2\int \frac{3 x + 1}{\left(x^{2} + 4 x\right) + 1}\, dx = C - 5 \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{3 \log{\left(x^{2} + 4 x + 1 \right)}}{2}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.9005
Trazar el gráfico f(x)
Respuesta [src] 
/        ___\                  /        ___\                  /        ___\                  /        ___\               
|3   5*\/ 3 |    /      ___\   |3   5*\/ 3 |    /      ___\   |3   5*\/ 3 |    /      ___\   |3   5*\/ 3 |    /      ___\
|- - -------|*log\3 - \/ 3 / + |- + -------|*log\3 + \/ 3 / - |- - -------|*log\2 - \/ 3 / - |- + -------|*log\2 + \/ 3 /
\2      6   /                  \2      6   /                  \2      6   /                  \2      6   /
−(536+32)log⁡(3+2)+(32−536)log⁡(3−3)−(32−536)log⁡(2−3)+(536+32)log⁡(3+3)- \left(\frac{5 \sqrt{3}}{6} + \frac{3}{2}\right) \log{\left(\sqrt{3} + 2 \right)} + \left(\frac{3}{2} - \frac{5 \sqrt{3}}{6}\right) \log{\left(3 - \sqrt{3} \right)} - \left(\frac{3}{2} - \frac{5 \sqrt{3}}{6}\right) \log{\left(2 - \sqrt{3} \right)} + \left(\frac{5 \sqrt{3}}{6} + \frac{3}{2}\right) \log{\left(\sqrt{3} + 3 \right)} 
=
= 
/        ___\                  /        ___\                  /        ___\                  /        ___\               
|3   5*\/ 3 |    /      ___\   |3   5*\/ 3 |    /      ___\   |3   5*\/ 3 |    /      ___\   |3   5*\/ 3 |    /      ___\
|- - -------|*log\3 - \/ 3 / + |- + -------|*log\3 + \/ 3 / - |- - -------|*log\2 - \/ 3 / - |- + -------|*log\2 + \/ 3 /
\2      6   /                  \2      6   /                  \2      6   /                  \2      6   /
−(536+32)log⁡(3+2)+(32−536)log⁡(3−3)−(32−536)log⁡(2−3)+(536+32)log⁡(3+3)- \left(\frac{5 \sqrt{3}}{6} + \frac{3}{2}\right) \log{\left(\sqrt{3} + 2 \right)} + \left(\frac{3}{2} - \frac{5 \sqrt{3}}{6}\right) \log{\left(3 - \sqrt{3} \right)} - \left(\frac{3}{2} - \frac{5 \sqrt{3}}{6}\right) \log{\left(2 - \sqrt{3} \right)} + \left(\frac{5 \sqrt{3}}{6} + \frac{3}{2}\right) \log{\left(\sqrt{3} + 3 \right)} 
(3/2 - 5*sqrt(3)/6)*log(3 - sqrt(3)) + (3/2 + 5*sqrt(3)/6)*log(3 + sqrt(3)) - (3/2 - 5*sqrt(3)/6)*log(2 - sqrt(3)) - (3/2 + 5*sqrt(3)/6)*log(2 + sqrt(3))
Respuesta numérica [src] 
0.786774213089717
0.786774213089717

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

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