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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                
  /                
 |                 
 |     x + 2       
 |  ------------ dx
 |   2             
 |  x  + 3*x + 1   
 |                 
/                  
0
∫01x+2(x2+3x)+1 dx\int\limits_{0}^{1} \frac{x + 2}{\left(x^{2} + 3 x\right) + 1}\, dx 
Integral((x + 2)/(x^2 + 3*x + 1), (x, 0, 1))
Respuesta (Indefinida) [src] 
                                               //            /    ___          \                       \
                                               ||   ___      |2*\/ 5 *(3/2 + x)|                       |
                                               ||-\/ 5 *acoth|-----------------|                       |
  /                                            ||            \        5        /                2      |
 |                          /     2      \     ||--------------------------------  for (3/2 + x)  > 5/4|
 |    x + 2              log\1 + x  + 3*x/     ||               10                                     |
 | ------------ dx = C + ----------------- + 2*|<                                                      |
 |  2                            2             ||            /    ___          \                       |
 | x  + 3*x + 1                                ||   ___      |2*\/ 5 *(3/2 + x)|                       |
 |                                             ||-\/ 5 *atanh|-----------------|                       |
/                                              ||            \        5        /                2      |
                                               ||--------------------------------  for (3/2 + x)  < 5/4|
                                               \\               10                                     /
∫x+2(x2+3x)+1 dx=C+2({−5acoth⁡(25(x+32)5)10for (x+32)2>54−5atanh⁡(25(x+32)5)10for (x+32)2<54)+log⁡(x2+3x+1)2\int \frac{x + 2}{\left(x^{2} + 3 x\right) + 1}\, dx = C + 2 \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(x^{2} + 3 x + 1 \right)}}{2}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.902.5-2.5
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     10 /    \2     2  /   \2     10 /    \2     2  /   \2     10 /    \2     2  /   \2     10 /    \2     2  /
−(12−510)log⁡(52+32)+(510+12)log⁡(52−52)+(12−510)log⁡(52+52)−(510+12)log⁡(32−52)- \left(\frac{1}{2} - \frac{\sqrt{5}}{10}\right) \log{\left(\frac{\sqrt{5}}{2} + \frac{3}{2} \right)} + \left(\frac{\sqrt{5}}{10} + \frac{1}{2}\right) \log{\left(\frac{5}{2} - \frac{\sqrt{5}}{2} \right)} + \left(\frac{1}{2} - \frac{\sqrt{5}}{10}\right) \log{\left(\frac{\sqrt{5}}{2} + \frac{5}{2} \right)} - \left(\frac{\sqrt{5}}{10} + \frac{1}{2}\right) \log{\left(\frac{3}{2} - \frac{\sqrt{5}}{2} \right)} 
=
= 
/      ___\    /      ___\   /      ___\    /      ___\   /      ___\    /      ___\   /      ___\    /      ___\
|1   \/ 5 |    |5   \/ 5 |   |1   \/ 5 |    |5   \/ 5 |   |1   \/ 5 |    |3   \/ 5 |   |1   \/ 5 |    |3   \/ 5 |
|- - -----|*log|- + -----| + |- + -----|*log|- - -----| - |- - -----|*log|- + -----| - |- + -----|*log|- - -----|
\2     10 /    \2     2  /   \2     10 /    \2     2  /   \2     10 /    \2     2  /   \2     10 /    \2     2  /
−(12−510)log⁡(52+32)+(510+12)log⁡(52−52)+(12−510)log⁡(52+52)−(510+12)log⁡(32−52)- \left(\frac{1}{2} - \frac{\sqrt{5}}{10}\right) \log{\left(\frac{\sqrt{5}}{2} + \frac{3}{2} \right)} + \left(\frac{\sqrt{5}}{10} + \frac{1}{2}\right) \log{\left(\frac{5}{2} - \frac{\sqrt{5}}{2} \right)} + \left(\frac{1}{2} - \frac{\sqrt{5}}{10}\right) \log{\left(\frac{\sqrt{5}}{2} + \frac{5}{2} \right)} - \left(\frac{\sqrt{5}}{10} + \frac{1}{2}\right) \log{\left(\frac{3}{2} - \frac{\sqrt{5}}{2} \right)} 
(1/2 - sqrt(5)/10)*log(5/2 + sqrt(5)/2) + (1/2 + sqrt(5)/10)*log(5/2 - sqrt(5)/2) - (1/2 - sqrt(5)/10)*log(3/2 + sqrt(5)/2) - (1/2 + sqrt(5)/10)*log(3/2 - sqrt(5)/2)
Respuesta numérica [src] 
1.01992342669905
1.01992342669905

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

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