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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                  
  /                  
 |                   
 |     1 + 2*x       
 |  -------------- dx
 |     2             
 |  2*x  + 6*x + 2   
 |                   
/                    
0
$$\int\limits_{0}^{1} \frac{2 x + 1}{\left(2 x^{2} + 6 x\right) + 2}\, dx$$ 
Integral((1 + 2*x)/(2*x^2 + 6*x + 2), (x, 0, 1))
Respuesta (Indefinida) [src] 
                                                   //            /    ___          \                       \
                                                   ||   ___      |2*\/ 5 *(3/2 + x)|                       |
                                                   ||-\/ 5 *acoth|-----------------|                       |
  /                                                ||            \        5        /                2      |
 |                            /       2      \     ||--------------------------------  for (3/2 + x)  > 5/4|
 |    1 + 2*x              log\2 + 2*x  + 6*x/     ||               10                                     |
 | -------------- dx = C + ------------------- - 4*|<                                                      |
 |    2                             2              ||            /    ___          \                       |
 | 2*x  + 6*x + 2                                  ||   ___      |2*\/ 5 *(3/2 + x)|                       |
 |                                                 ||-\/ 5 *atanh|-----------------|                       |
/                                                  ||            \        5        /                2      |
                                                   ||--------------------------------  for (3/2 + x)  < 5/4|
                                                   \\               10                                     /
$$\int \frac{2 x + 1}{\left(2 x^{2} + 6 x\right) + 2}\, dx = C - 4 \left(\begin{cases} - \frac{\sqrt{5} \operatorname{acoth}{\left(\frac{2 \sqrt{5} \left(x + \frac{3}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(x + \frac{3}{2}\right)^{2} > \frac{5}{4} \\- \frac{\sqrt{5} \operatorname{atanh}{\left(\frac{2 \sqrt{5} \left(x + \frac{3}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(x + \frac{3}{2}\right)^{2} < \frac{5}{4} \end{cases}\right) + \frac{\log{\left(2 x^{2} + 6 x + 2 \right)}}{2}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
/      ___\    /      ___\   /      ___\    /      ___\   /      ___\    /      ___\   /      ___\    /      ___\
|1   \/ 5 |    |5   \/ 5 |   |1   \/ 5 |    |5   \/ 5 |   |1   \/ 5 |    |3   \/ 5 |   |1   \/ 5 |    |3   \/ 5 |
|- - -----|*log|- - -----| + |- + -----|*log|- + -----| - |- - -----|*log|- - -----| - |- + -----|*log|- + -----|
\2     5  /    \2     2  /   \2     5  /    \2     2  /   \2     5  /    \2     2  /   \2     5  /    \2     2  /
$$- \left(\frac{\sqrt{5}}{5} + \frac{1}{2}\right) \log{\left(\frac{\sqrt{5}}{2} + \frac{3}{2} \right)} + \left(\frac{1}{2} - \frac{\sqrt{5}}{5}\right) \log{\left(\frac{5}{2} - \frac{\sqrt{5}}{2} \right)} - \left(\frac{1}{2} - \frac{\sqrt{5}}{5}\right) \log{\left(\frac{3}{2} - \frac{\sqrt{5}}{2} \right)} + \left(\frac{\sqrt{5}}{5} + \frac{1}{2}\right) \log{\left(\frac{\sqrt{5}}{2} + \frac{5}{2} \right)}$$ 
=
= 
/      ___\    /      ___\   /      ___\    /      ___\   /      ___\    /      ___\   /      ___\    /      ___\
|1   \/ 5 |    |5   \/ 5 |   |1   \/ 5 |    |5   \/ 5 |   |1   \/ 5 |    |3   \/ 5 |   |1   \/ 5 |    |3   \/ 5 |
|- - -----|*log|- - -----| + |- + -----|*log|- + -----| - |- - -----|*log|- - -----| - |- + -----|*log|- + -----|
\2     5  /    \2     2  /   \2     5  /    \2     2  /   \2     5  /    \2     2  /   \2     5  /    \2     2  /
$$- \left(\frac{\sqrt{5}}{5} + \frac{1}{2}\right) \log{\left(\frac{\sqrt{5}}{2} + \frac{3}{2} \right)} + \left(\frac{1}{2} - \frac{\sqrt{5}}{5}\right) \log{\left(\frac{5}{2} - \frac{\sqrt{5}}{2} \right)} - \left(\frac{1}{2} - \frac{\sqrt{5}}{5}\right) \log{\left(\frac{3}{2} - \frac{\sqrt{5}}{2} \right)} + \left(\frac{\sqrt{5}}{5} + \frac{1}{2}\right) \log{\left(\frac{\sqrt{5}}{2} + \frac{5}{2} \right)}$$ 
(1/2 - sqrt(5)/5)*log(5/2 - sqrt(5)/2) + (1/2 + sqrt(5)/5)*log(5/2 + sqrt(5)/2) - (1/2 - sqrt(5)/5)*log(3/2 - sqrt(5)/2) - (1/2 + sqrt(5)/5)*log(3/2 + sqrt(5)/2)
Respuesta numérica [src] 
0.374310015253046
0.374310015253046

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

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