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Resolución de integrales/ x^2+5/ (3x-2)/ (3x-2)/(x^2+5x+1)

Integral de (3x-2)/(x^2+5x+1) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                
  /                
 |                 
 |    3*x - 2      
 |  ------------ dx
 |   2             
 |  x  + 5*x + 1   
 |                 
/                  
0
$$\int\limits_{0}^{1} \frac{3 x - 2}{\left(x^{2} + 5 x\right) + 1}\, dx$$ 
Integral((3*x - 2)/(x^2 + 5*x + 1), (x, 0, 1))
Respuesta (Indefinida) [src] 
                            //             /    ____          \                        \                      
                            ||   ____      |2*\/ 21 *(5/2 + x)|                        |                      
                            ||-\/ 21 *acoth|------------------|                        |                      
  /                         ||             \        21        /                2       |                      
 |                          ||----------------------------------  for (5/2 + x)  > 21/4|        /     2      \
 |   3*x - 2                ||                42                                       |   3*log\1 + x  + 5*x/
 | ------------ dx = C - 38*|<                                                         | + -------------------
 |  2                       ||             /    ____          \                        |            2         
 | x  + 5*x + 1             ||   ____      |2*\/ 21 *(5/2 + x)|                        |                      
 |                          ||-\/ 21 *atanh|------------------|                        |                      
/                           ||             \        21        /                2       |                      
                            ||----------------------------------  for (5/2 + x)  < 21/4|                      
                            \\                42                                       /
$$\int \frac{3 x - 2}{\left(x^{2} + 5 x\right) + 1}\, dx = C - 38 \left(\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left(\frac{2 \sqrt{21} \left(x + \frac{5}{2}\right)}{21} \right)}}{42} & \text{for}\: \left(x + \frac{5}{2}\right)^{2} > \frac{21}{4} \\- \frac{\sqrt{21} \operatorname{atanh}{\left(\frac{2 \sqrt{21} \left(x + \frac{5}{2}\right)}{21} \right)}}{42} & \text{for}\: \left(x + \frac{5}{2}\right)^{2} < \frac{21}{4} \end{cases}\right) + \frac{3 \log{\left(x^{2} + 5 x + 1 \right)}}{2}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
/         ____\    /      ____\   /         ____\    /      ____\   /         ____\    /      ____\   /         ____\    /      ____\
|3   19*\/ 21 |    |7   \/ 21 |   |3   19*\/ 21 |    |7   \/ 21 |   |3   19*\/ 21 |    |5   \/ 21 |   |3   19*\/ 21 |    |5   \/ 21 |
|- - ---------|*log|- - ------| + |- + ---------|*log|- + ------| - |- - ---------|*log|- - ------| - |- + ---------|*log|- + ------|
\2       42   /    \2     2   /   \2       42   /    \2     2   /   \2       42   /    \2     2   /   \2       42   /    \2     2   /
$$- \left(\frac{3}{2} + \frac{19 \sqrt{21}}{42}\right) \log{\left(\frac{\sqrt{21}}{2} + \frac{5}{2} \right)} - \left(\frac{3}{2} - \frac{19 \sqrt{21}}{42}\right) \log{\left(\frac{5}{2} - \frac{\sqrt{21}}{2} \right)} + \left(\frac{3}{2} - \frac{19 \sqrt{21}}{42}\right) \log{\left(\frac{7}{2} - \frac{\sqrt{21}}{2} \right)} + \left(\frac{3}{2} + \frac{19 \sqrt{21}}{42}\right) \log{\left(\frac{\sqrt{21}}{2} + \frac{7}{2} \right)}$$ 
=
= 
/         ____\    /      ____\   /         ____\    /      ____\   /         ____\    /      ____\   /         ____\    /      ____\
|3   19*\/ 21 |    |7   \/ 21 |   |3   19*\/ 21 |    |7   \/ 21 |   |3   19*\/ 21 |    |5   \/ 21 |   |3   19*\/ 21 |    |5   \/ 21 |
|- - ---------|*log|- - ------| + |- + ---------|*log|- + ------| - |- - ---------|*log|- - ------| - |- + ---------|*log|- + ------|
\2       42   /    \2     2   /   \2       42   /    \2     2   /   \2       42   /    \2     2   /   \2       42   /    \2     2   /
$$- \left(\frac{3}{2} + \frac{19 \sqrt{21}}{42}\right) \log{\left(\frac{\sqrt{21}}{2} + \frac{5}{2} \right)} - \left(\frac{3}{2} - \frac{19 \sqrt{21}}{42}\right) \log{\left(\frac{5}{2} - \frac{\sqrt{21}}{2} \right)} + \left(\frac{3}{2} - \frac{19 \sqrt{21}}{42}\right) \log{\left(\frac{7}{2} - \frac{\sqrt{21}}{2} \right)} + \left(\frac{3}{2} + \frac{19 \sqrt{21}}{42}\right) \log{\left(\frac{\sqrt{21}}{2} + \frac{7}{2} \right)}$$ 
(3/2 - 19*sqrt(21)/42)*log(7/2 - sqrt(21)/2) + (3/2 + 19*sqrt(21)/42)*log(7/2 + sqrt(21)/2) - (3/2 - 19*sqrt(21)/42)*log(5/2 - sqrt(21)/2) - (3/2 + 19*sqrt(21)/42)*log(5/2 + sqrt(21)/2)
Respuesta numérica [src] 
-0.329219203614111
-0.329219203614111

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

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