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Resolución de integrales/ 12x+1/ x^(2)/ (3x-1)/ (3x-1)/(3x^(2)+12x+1)

Integral de (3x-1)/(3x^(2)+12x+1) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                   
  /                   
 |                    
 |      3*x - 1       
 |  --------------- dx
 |     2              
 |  3*x  + 12*x + 1   
 |                    
/                     
0
∫013x−1(3x2+12x)+1 dx\int\limits_{0}^{1} \frac{3 x - 1}{\left(3 x^{2} + 12 x\right) + 1}\, dx 
Integral((3*x - 1)/(3*x^2 + 12*x + 1), (x, 0, 1))
Respuesta (Indefinida) [src] 
                                                     //             /  ____        \                      \
                                                     ||   ____      |\/ 33 *(2 + x)|                      |
                                                     ||-\/ 33 *acoth|--------------|                      |
  /                                                  ||             \      11      /              2       |
 |                             /       2       \     ||------------------------------  for (2 + x)  > 11/3|
 |     3*x - 1              log\1 + 3*x  + 12*x/     ||              33                                   |
 | --------------- dx = C + -------------------- - 7*|<                                                   |
 |    2                              2               ||             /  ____        \                      |
 | 3*x  + 12*x + 1                                   ||   ____      |\/ 33 *(2 + x)|                      |
 |                                                   ||-\/ 33 *atanh|--------------|                      |
/                                                    ||             \      11      /              2       |
                                                     ||------------------------------  for (2 + x)  < 11/3|
                                                     \\              33                                   /
∫3x−1(3x2+12x)+1 dx=C−7({−33acoth⁡(33(x+2)11)33for (x+2)2>113−33atanh⁡(33(x+2)11)33for (x+2)2<113)+log⁡(3x2+12x+1)2\int \frac{3 x - 1}{\left(3 x^{2} + 12 x\right) + 1}\, dx = C - 7 \left(\begin{cases} - \frac{\sqrt{33} \operatorname{acoth}{\left(\frac{\sqrt{33} \left(x + 2\right)}{11} \right)}}{33} & \text{for}\: \left(x + 2\right)^{2} > \frac{11}{3} \\- \frac{\sqrt{33} \operatorname{atanh}{\left(\frac{\sqrt{33} \left(x + 2\right)}{11} \right)}}{33} & \text{for}\: \left(x + 2\right)^{2} < \frac{11}{3} \end{cases}\right) + \frac{\log{\left(3 x^{2} + 12 x + 1 \right)}}{2}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.905-5
Trazar el gráfico f(x)
Respuesta [src] 
/        ____\    /      ____\   /        ____\    /      ____\   /        ____\    /      ____\   /        ____\    /      ____\
|1   7*\/ 33 |    |    \/ 33 |   |1   7*\/ 33 |    |    \/ 33 |   |1   7*\/ 33 |    |    \/ 33 |   |1   7*\/ 33 |    |    \/ 33 |
|- - --------|*log|3 - ------| + |- + --------|*log|3 + ------| - |- - --------|*log|2 - ------| - |- + --------|*log|2 + ------|
\2      66   /    \      3   /   \2      66   /    \      3   /   \2      66   /    \      3   /   \2      66   /    \      3   /
−(12+73366)log⁡(333+2)−(12−73366)log⁡(2−333)+(12−73366)log⁡(3−333)+(12+73366)log⁡(333+3)- \left(\frac{1}{2} + \frac{7 \sqrt{33}}{66}\right) \log{\left(\frac{\sqrt{33}}{3} + 2 \right)} - \left(\frac{1}{2} - \frac{7 \sqrt{33}}{66}\right) \log{\left(2 - \frac{\sqrt{33}}{3} \right)} + \left(\frac{1}{2} - \frac{7 \sqrt{33}}{66}\right) \log{\left(3 - \frac{\sqrt{33}}{3} \right)} + \left(\frac{1}{2} + \frac{7 \sqrt{33}}{66}\right) \log{\left(\frac{\sqrt{33}}{3} + 3 \right)} 
=
= 
/        ____\    /      ____\   /        ____\    /      ____\   /        ____\    /      ____\   /        ____\    /      ____\
|1   7*\/ 33 |    |    \/ 33 |   |1   7*\/ 33 |    |    \/ 33 |   |1   7*\/ 33 |    |    \/ 33 |   |1   7*\/ 33 |    |    \/ 33 |
|- - --------|*log|3 - ------| + |- + --------|*log|3 + ------| - |- - --------|*log|2 - ------| - |- + --------|*log|2 + ------|
\2      66   /    \      3   /   \2      66   /    \      3   /   \2      66   /    \      3   /   \2      66   /    \      3   /
−(12+73366)log⁡(333+2)−(12−73366)log⁡(2−333)+(12−73366)log⁡(3−333)+(12+73366)log⁡(333+3)- \left(\frac{1}{2} + \frac{7 \sqrt{33}}{66}\right) \log{\left(\frac{\sqrt{33}}{3} + 2 \right)} - \left(\frac{1}{2} - \frac{7 \sqrt{33}}{66}\right) \log{\left(2 - \frac{\sqrt{33}}{3} \right)} + \left(\frac{1}{2} - \frac{7 \sqrt{33}}{66}\right) \log{\left(3 - \frac{\sqrt{33}}{3} \right)} + \left(\frac{1}{2} + \frac{7 \sqrt{33}}{66}\right) \log{\left(\frac{\sqrt{33}}{3} + 3 \right)} 
(1/2 - 7*sqrt(33)/66)*log(3 - sqrt(33)/3) + (1/2 + 7*sqrt(33)/66)*log(3 + sqrt(33)/3) - (1/2 - 7*sqrt(33)/66)*log(2 - sqrt(33)/3) - (1/2 + 7*sqrt(33)/66)*log(2 + sqrt(33)/3)
Respuesta numérica [src] 
-0.0257665775658668
-0.0257665775658668

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

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