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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                
  /                
 |                 
 |       1         
 |  ------------ dx
 |   2             
 |  x  + 3*x + 1   
 |                 
/                  
0
$$\int\limits_{0}^{1} \frac{1}{\left(x^{2} + 3 x\right) + 1}\, dx$$ 
Integral(1/(x^2 + 3*x + 1), (x, 0, 1))
Respuesta (Indefinida) [src] 
                           //            /    ___          \                       \
                           ||   ___      |2*\/ 5 *(3/2 + x)|                       |
                           ||-\/ 5 *acoth|-----------------|                       |
  /                        ||            \        5        /                2      |
 |                         ||--------------------------------  for (3/2 + x)  > 5/4|
 |      1                  ||               10                                     |
 | ------------ dx = C + 4*|<                                                      |
 |  2                      ||            /    ___          \                       |
 | x  + 3*x + 1            ||   ___      |2*\/ 5 *(3/2 + x)|                       |
 |                         ||-\/ 5 *atanh|-----------------|                       |
/                          ||            \        5        /                2      |
                           ||--------------------------------  for (3/2 + x)  < 5/4|
                           \\               10                                     /
$$\int \frac{1}{\left(x^{2} + 3 x\right) + 1}\, dx = C + 4 \left(\begin{cases} - \frac{\sqrt{5} \operatorname{acoth}{\left(\frac{2 \sqrt{5} \left(x + \frac{3}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(x + \frac{3}{2}\right)^{2} > \frac{5}{4} \\- \frac{\sqrt{5} \operatorname{atanh}{\left(\frac{2 \sqrt{5} \left(x + \frac{3}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(x + \frac{3}{2}\right)^{2} < \frac{5}{4} \end{cases}\right)$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
           /      ___\            /      ___\            /      ___\            /      ___\
    ___    |3   \/ 5 |     ___    |5   \/ 5 |     ___    |3   \/ 5 |     ___    |5   \/ 5 |
  \/ 5 *log|- - -----|   \/ 5 *log|- + -----|   \/ 5 *log|- + -----|   \/ 5 *log|- - -----|
           \2     2  /            \2     2  /            \2     2  /            \2     2  /
- -------------------- - -------------------- + -------------------- + --------------------
           5                      5                      5                      5
$$- \frac{\sqrt{5} \log{\left(\frac{\sqrt{5}}{2} + \frac{5}{2} \right)}}{5} + \frac{\sqrt{5} \log{\left(\frac{5}{2} - \frac{\sqrt{5}}{2} \right)}}{5} - \frac{\sqrt{5} \log{\left(\frac{3}{2} - \frac{\sqrt{5}}{2} \right)}}{5} + \frac{\sqrt{5} \log{\left(\frac{\sqrt{5}}{2} + \frac{3}{2} \right)}}{5}$$ 
=
= 
           /      ___\            /      ___\            /      ___\            /      ___\
    ___    |3   \/ 5 |     ___    |5   \/ 5 |     ___    |3   \/ 5 |     ___    |5   \/ 5 |
  \/ 5 *log|- - -----|   \/ 5 *log|- + -----|   \/ 5 *log|- + -----|   \/ 5 *log|- - -----|
           \2     2  /            \2     2  /            \2     2  /            \2     2  /
- -------------------- - -------------------- + -------------------- + --------------------
           5                      5                      5                      5
$$- \frac{\sqrt{5} \log{\left(\frac{\sqrt{5}}{2} + \frac{5}{2} \right)}}{5} + \frac{\sqrt{5} \log{\left(\frac{5}{2} - \frac{\sqrt{5}}{2} \right)}}{5} - \frac{\sqrt{5} \log{\left(\frac{3}{2} - \frac{\sqrt{5}}{2} \right)}}{5} + \frac{\sqrt{5} \log{\left(\frac{\sqrt{5}}{2} + \frac{3}{2} \right)}}{5}$$ 
-sqrt(5)*log(3/2 - sqrt(5)/2)/5 - sqrt(5)*log(5/2 + sqrt(5)/2)/5 + sqrt(5)*log(3/2 + sqrt(5)/2)/5 + sqrt(5)*log(5/2 - sqrt(5)/2)/5
Respuesta numérica [src] 
0.430408940964004
0.430408940964004

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

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