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Resolución de integrales/ (3x-1)/ x^2+1/ (3x-1)/(x^2+10x+6)

Integral de (3x-1)/(x^2+10x+6) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                 
  /                 
 |                  
 |     3*x - 1      
 |  ------------- dx
 |   2              
 |  x  + 10*x + 6   
 |                  
/                   
0
$$\int\limits_{0}^{1} \frac{3 x - 1}{\left(x^{2} + 10 x\right) + 6}\, dx$$ 
Integral((3*x - 1)/(x^2 + 10*x + 6), (x, 0, 1))
Respuesta (Indefinida) [src] 
                             //             /  ____        \                    \                       
                             ||   ____      |\/ 19 *(5 + x)|                    |                       
                             ||-\/ 19 *acoth|--------------|                    |                       
  /                          ||             \      19      /              2     |                       
 |                           ||------------------------------  for (5 + x)  > 19|        /     2       \
 |    3*x - 1                ||              19                                 |   3*log\6 + x  + 10*x/
 | ------------- dx = C - 16*|<                                                 | + --------------------
 |  2                        ||             /  ____        \                    |            2          
 | x  + 10*x + 6             ||   ____      |\/ 19 *(5 + x)|                    |                       
 |                           ||-\/ 19 *atanh|--------------|                    |                       
/                            ||             \      19      /              2     |                       
                             ||------------------------------  for (5 + x)  < 19|                       
                             \\              19                                 /
$$\int \frac{3 x - 1}{\left(x^{2} + 10 x\right) + 6}\, dx = C - 16 \left(\begin{cases} - \frac{\sqrt{19} \operatorname{acoth}{\left(\frac{\sqrt{19} \left(x + 5\right)}{19} \right)}}{19} & \text{for}\: \left(x + 5\right)^{2} > 19 \\- \frac{\sqrt{19} \operatorname{atanh}{\left(\frac{\sqrt{19} \left(x + 5\right)}{19} \right)}}{19} & \text{for}\: \left(x + 5\right)^{2} < 19 \end{cases}\right) + \frac{3 \log{\left(x^{2} + 10 x + 6 \right)}}{2}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
/        ____\                   /        ____\                   /        ____\                   /        ____\                
|3   8*\/ 19 |    /      ____\   |3   8*\/ 19 |    /      ____\   |3   8*\/ 19 |    /      ____\   |3   8*\/ 19 |    /      ____\
|- - --------|*log\6 - \/ 19 / + |- + --------|*log\6 + \/ 19 / - |- - --------|*log\5 - \/ 19 / - |- + --------|*log\5 + \/ 19 /
\2      19   /                   \2      19   /                   \2      19   /                   \2      19   /
$$- \left(\frac{3}{2} + \frac{8 \sqrt{19}}{19}\right) \log{\left(\sqrt{19} + 5 \right)} + \left(\frac{3}{2} - \frac{8 \sqrt{19}}{19}\right) \log{\left(6 - \sqrt{19} \right)} - \left(\frac{3}{2} - \frac{8 \sqrt{19}}{19}\right) \log{\left(5 - \sqrt{19} \right)} + \left(\frac{3}{2} + \frac{8 \sqrt{19}}{19}\right) \log{\left(\sqrt{19} + 6 \right)}$$ 
=
= 
/        ____\                   /        ____\                   /        ____\                   /        ____\                
|3   8*\/ 19 |    /      ____\   |3   8*\/ 19 |    /      ____\   |3   8*\/ 19 |    /      ____\   |3   8*\/ 19 |    /      ____\
|- - --------|*log\6 - \/ 19 / + |- + --------|*log\6 + \/ 19 / - |- - --------|*log\5 - \/ 19 / - |- + --------|*log\5 + \/ 19 /
\2      19   /                   \2      19   /                   \2      19   /                   \2      19   /
$$- \left(\frac{3}{2} + \frac{8 \sqrt{19}}{19}\right) \log{\left(\sqrt{19} + 5 \right)} + \left(\frac{3}{2} - \frac{8 \sqrt{19}}{19}\right) \log{\left(6 - \sqrt{19} \right)} - \left(\frac{3}{2} - \frac{8 \sqrt{19}}{19}\right) \log{\left(5 - \sqrt{19} \right)} + \left(\frac{3}{2} + \frac{8 \sqrt{19}}{19}\right) \log{\left(\sqrt{19} + 6 \right)}$$ 
(3/2 - 8*sqrt(19)/19)*log(6 - sqrt(19)) + (3/2 + 8*sqrt(19)/19)*log(6 + sqrt(19)) - (3/2 - 8*sqrt(19)/19)*log(5 - sqrt(19)) - (3/2 + 8*sqrt(19)/19)*log(5 + sqrt(19))
Respuesta numérica [src] 
0.0234119054163882
0.0234119054163882

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

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