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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
  1                
  /                
 |                 
 |       1         
 |  ------------ dx
 |   2             
 |  x  + 4*x + 1   
 |                 
/                  
0                  
$$\int\limits_{0}^{1} \frac{1}{\left(x^{2} + 4 x\right) + 1}\, dx$$
Integral(1/(x^2 + 4*x + 1), (x, 0, 1))
Respuesta (Indefinida) [src]
                         //            /  ___        \                   \
                         ||   ___      |\/ 3 *(2 + x)|                   |
                         ||-\/ 3 *acoth|-------------|                   |
  /                      ||            \      3      /              2    |
 |                       ||----------------------------  for (2 + x)  > 3|
 |      1                ||             3                                |
 | ------------ dx = C + |<                                              |
 |  2                    ||            /  ___        \                   |
 | x  + 4*x + 1          ||   ___      |\/ 3 *(2 + x)|                   |
 |                       ||-\/ 3 *atanh|-------------|                   |
/                        ||            \      3      /              2    |
                         ||----------------------------  for (2 + x)  < 3|
                         \\             3                                /
$$\int \frac{1}{\left(x^{2} + 4 x\right) + 1}\, dx = C + \begin{cases} - \frac{\sqrt{3} \operatorname{acoth}{\left(\frac{\sqrt{3} \left(x + 2\right)}{3} \right)}}{3} & \text{for}\: \left(x + 2\right)^{2} > 3 \\- \frac{\sqrt{3} \operatorname{atanh}{\left(\frac{\sqrt{3} \left(x + 2\right)}{3} \right)}}{3} & \text{for}\: \left(x + 2\right)^{2} < 3 \end{cases}$$
Gráfica
Respuesta [src]
    ___    /      ___\     ___    /      ___\     ___    /      ___\     ___    /      ___\
  \/ 3 *log\2 - \/ 3 /   \/ 3 *log\3 + \/ 3 /   \/ 3 *log\2 + \/ 3 /   \/ 3 *log\3 - \/ 3 /
- -------------------- - -------------------- + -------------------- + --------------------
           6                      6                      6                      6          
$$- \frac{\sqrt{3} \log{\left(\sqrt{3} + 3 \right)}}{6} + \frac{\sqrt{3} \log{\left(3 - \sqrt{3} \right)}}{6} - \frac{\sqrt{3} \log{\left(2 - \sqrt{3} \right)}}{6} + \frac{\sqrt{3} \log{\left(\sqrt{3} + 2 \right)}}{6}$$
=
=
    ___    /      ___\     ___    /      ___\     ___    /      ___\     ___    /      ___\
  \/ 3 *log\2 - \/ 3 /   \/ 3 *log\3 + \/ 3 /   \/ 3 *log\2 + \/ 3 /   \/ 3 *log\3 - \/ 3 /
- -------------------- - -------------------- + -------------------- + --------------------
           6                      6                      6                      6          
$$- \frac{\sqrt{3} \log{\left(\sqrt{3} + 3 \right)}}{6} + \frac{\sqrt{3} \log{\left(3 - \sqrt{3} \right)}}{6} - \frac{\sqrt{3} \log{\left(2 - \sqrt{3} \right)}}{6} + \frac{\sqrt{3} \log{\left(\sqrt{3} + 2 \right)}}{6}$$
-sqrt(3)*log(2 - sqrt(3))/6 - sqrt(3)*log(3 + sqrt(3))/6 + sqrt(3)*log(2 + sqrt(3))/6 + sqrt(3)*log(3 - sqrt(3))/6
Respuesta numérica [src]
0.380172998150473
0.380172998150473

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