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Resolución de integrales/ (x+3)/ 3x^2+6x/ (x+3)/(3x^2+6x+1)

Integral de (x+3)/(3x^2+6x+1) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                  
  /                  
 |                   
 |      x + 3        
 |  -------------- dx
 |     2             
 |  3*x  + 6*x + 1   
 |                   
/                    
0
∫01x+3(3x2+6x)+1 dx\int\limits_{0}^{1} \frac{x + 3}{\left(3 x^{2} + 6 x\right) + 1}\, dx 
Integral((x + 3)/(3*x^2 + 6*x + 1), (x, 0, 1))
Respuesta (Indefinida) [src] 
                             //            /  ___        \                     \                      
                             ||   ___      |\/ 6 *(1 + x)|                     |                      
                             ||-\/ 6 *acoth|-------------|                     |                      
  /                          ||            \      2      /              2      |                      
 |                           ||----------------------------  for (1 + x)  > 2/3|      /       2      \
 |     x + 3                 ||             6                                  |   log\1 + 3*x  + 6*x/
 | -------------- dx = C + 2*|<                                                | + -------------------
 |    2                      ||            /  ___        \                     |            6         
 | 3*x  + 6*x + 1            ||   ___      |\/ 6 *(1 + x)|                     |                      
 |                           ||-\/ 6 *atanh|-------------|                     |                      
/                            ||            \      2      /              2      |                      
                             ||----------------------------  for (1 + x)  < 2/3|                      
                             \\             6                                  /
∫x+3(3x2+6x)+1 dx=C+2({−6acoth⁡(6(x+1)2)6for (x+1)2>23−6atanh⁡(6(x+1)2)6for (x+1)2<23)+log⁡(3x2+6x+1)6\int \frac{x + 3}{\left(3 x^{2} + 6 x\right) + 1}\, dx = C + 2 \left(\begin{cases} - \frac{\sqrt{6} \operatorname{acoth}{\left(\frac{\sqrt{6} \left(x + 1\right)}{2} \right)}}{6} & \text{for}\: \left(x + 1\right)^{2} > \frac{2}{3} \\- \frac{\sqrt{6} \operatorname{atanh}{\left(\frac{\sqrt{6} \left(x + 1\right)}{2} \right)}}{6} & \text{for}\: \left(x + 1\right)^{2} < \frac{2}{3} \end{cases}\right) + \frac{\log{\left(3 x^{2} + 6 x + 1 \right)}}{6}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.905-5
Trazar el gráfico f(x)
Respuesta [src] 
/      ___\    /      ___\   /      ___\    /      ___\   /      ___\    /      ___\   /      ___\    /      ___\
|1   \/ 6 |    |    \/ 6 |   |1   \/ 6 |    |    \/ 6 |   |1   \/ 6 |    |    \/ 6 |   |1   \/ 6 |    |    \/ 6 |
|- - -----|*log|2 + -----| + |- + -----|*log|2 - -----| - |- - -----|*log|1 + -----| - |- + -----|*log|1 - -----|
\6     6  /    \      3  /   \6     6  /    \      3  /   \6     6  /    \      3  /   \6     6  /    \      3  /
(16−66)log⁡(63+2)+(16+66)log⁡(2−63)−(16−66)log⁡(63+1)−(16+66)log⁡(1−63)\left(\frac{1}{6} - \frac{\sqrt{6}}{6}\right) \log{\left(\frac{\sqrt{6}}{3} + 2 \right)} + \left(\frac{1}{6} + \frac{\sqrt{6}}{6}\right) \log{\left(2 - \frac{\sqrt{6}}{3} \right)} - \left(\frac{1}{6} - \frac{\sqrt{6}}{6}\right) \log{\left(\frac{\sqrt{6}}{3} + 1 \right)} - \left(\frac{1}{6} + \frac{\sqrt{6}}{6}\right) \log{\left(1 - \frac{\sqrt{6}}{3} \right)} 
=
= 
/      ___\    /      ___\   /      ___\    /      ___\   /      ___\    /      ___\   /      ___\    /      ___\
|1   \/ 6 |    |    \/ 6 |   |1   \/ 6 |    |    \/ 6 |   |1   \/ 6 |    |    \/ 6 |   |1   \/ 6 |    |    \/ 6 |
|- - -----|*log|2 + -----| + |- + -----|*log|2 - -----| - |- - -----|*log|1 + -----| - |- + -----|*log|1 - -----|
\6     6  /    \      3  /   \6     6  /    \      3  /   \6     6  /    \      3  /   \6     6  /    \      3  /
(16−66)log⁡(63+2)+(16+66)log⁡(2−63)−(16−66)log⁡(63+1)−(16+66)log⁡(1−63)\left(\frac{1}{6} - \frac{\sqrt{6}}{6}\right) \log{\left(\frac{\sqrt{6}}{3} + 2 \right)} + \left(\frac{1}{6} + \frac{\sqrt{6}}{6}\right) \log{\left(2 - \frac{\sqrt{6}}{3} \right)} - \left(\frac{1}{6} - \frac{\sqrt{6}}{6}\right) \log{\left(\frac{\sqrt{6}}{3} + 1 \right)} - \left(\frac{1}{6} + \frac{\sqrt{6}}{6}\right) \log{\left(1 - \frac{\sqrt{6}}{3} \right)} 
(1/6 - sqrt(6)/6)*log(2 + sqrt(6)/3) + (1/6 + sqrt(6)/6)*log(2 - sqrt(6)/3) - (1/6 - sqrt(6)/6)*log(1 + sqrt(6)/3) - (1/6 + sqrt(6)/6)*log(1 - sqrt(6)/3)
Respuesta numérica [src] 
0.965688212372477
0.965688212372477

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

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