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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                
  /                
 |                 
 |    3*x - 1      
 |  ------------ dx
 |   2             
 |  x  + 6*x + 3   
 |                 
/                  
0
∫013x−1(x2+6x)+3 dx\int\limits_{0}^{1} \frac{3 x - 1}{\left(x^{2} + 6 x\right) + 3}\, dx 
Integral((3*x - 1)/(x^2 + 6*x + 3), (x, 0, 1))
Respuesta (Indefinida) [src] 
                            //            /  ___        \                   \                      
                            ||   ___      |\/ 6 *(3 + x)|                   |                      
                            ||-\/ 6 *acoth|-------------|                   |                      
  /                         ||            \      6      /              2    |                      
 |                          ||----------------------------  for (3 + x)  > 6|        /     2      \
 |   3*x - 1                ||             6                                |   3*log\3 + x  + 6*x/
 | ------------ dx = C - 10*|<                                              | + -------------------
 |  2                       ||            /  ___        \                   |            2         
 | x  + 6*x + 3             ||   ___      |\/ 6 *(3 + x)|                   |                      
 |                          ||-\/ 6 *atanh|-------------|                   |                      
/                           ||            \      6      /              2    |                      
                            ||----------------------------  for (3 + x)  < 6|                      
                            \\             6                                /
∫3x−1(x2+6x)+3 dx=C−10({−6acoth⁡(6(x+3)6)6for (x+3)2>6−6atanh⁡(6(x+3)6)6for (x+3)2<6)+3log⁡(x2+6x+3)2\int \frac{3 x - 1}{\left(x^{2} + 6 x\right) + 3}\, dx = C - 10 \left(\begin{cases} - \frac{\sqrt{6} \operatorname{acoth}{\left(\frac{\sqrt{6} \left(x + 3\right)}{6} \right)}}{6} & \text{for}\: \left(x + 3\right)^{2} > 6 \\- \frac{\sqrt{6} \operatorname{atanh}{\left(\frac{\sqrt{6} \left(x + 3\right)}{6} \right)}}{6} & \text{for}\: \left(x + 3\right)^{2} < 6 \end{cases}\right) + \frac{3 \log{\left(x^{2} + 6 x + 3 \right)}}{2}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.90-510
Trazar el gráfico f(x)
Respuesta [src] 
/        ___\                  /        ___\                  /        ___\                  /        ___\               
|3   5*\/ 6 |    /      ___\   |3   5*\/ 6 |    /      ___\   |3   5*\/ 6 |    /      ___\   |3   5*\/ 6 |    /      ___\
|- - -------|*log\4 - \/ 6 / + |- + -------|*log\4 + \/ 6 / - |- - -------|*log\3 - \/ 6 / - |- + -------|*log\3 + \/ 6 /
\2      6   /                  \2      6   /                  \2      6   /                  \2      6   /
−(32+566)log⁡(6+3)−(32−566)log⁡(3−6)+(32−566)log⁡(4−6)+(32+566)log⁡(6+4)- \left(\frac{3}{2} + \frac{5 \sqrt{6}}{6}\right) \log{\left(\sqrt{6} + 3 \right)} - \left(\frac{3}{2} - \frac{5 \sqrt{6}}{6}\right) \log{\left(3 - \sqrt{6} \right)} + \left(\frac{3}{2} - \frac{5 \sqrt{6}}{6}\right) \log{\left(4 - \sqrt{6} \right)} + \left(\frac{3}{2} + \frac{5 \sqrt{6}}{6}\right) \log{\left(\sqrt{6} + 4 \right)} 
=
= 
/        ___\                  /        ___\                  /        ___\                  /        ___\               
|3   5*\/ 6 |    /      ___\   |3   5*\/ 6 |    /      ___\   |3   5*\/ 6 |    /      ___\   |3   5*\/ 6 |    /      ___\
|- - -------|*log\4 - \/ 6 / + |- + -------|*log\4 + \/ 6 / - |- - -------|*log\3 - \/ 6 / - |- + -------|*log\3 + \/ 6 /
\2      6   /                  \2      6   /                  \2      6   /                  \2      6   /
−(32+566)log⁡(6+3)−(32−566)log⁡(3−6)+(32−566)log⁡(4−6)+(32+566)log⁡(6+4)- \left(\frac{3}{2} + \frac{5 \sqrt{6}}{6}\right) \log{\left(\sqrt{6} + 3 \right)} - \left(\frac{3}{2} - \frac{5 \sqrt{6}}{6}\right) \log{\left(3 - \sqrt{6} \right)} + \left(\frac{3}{2} - \frac{5 \sqrt{6}}{6}\right) \log{\left(4 - \sqrt{6} \right)} + \left(\frac{3}{2} + \frac{5 \sqrt{6}}{6}\right) \log{\left(\sqrt{6} + 4 \right)} 
(3/2 - 5*sqrt(6)/6)*log(4 - sqrt(6)) + (3/2 + 5*sqrt(6)/6)*log(4 + sqrt(6)) - (3/2 - 5*sqrt(6)/6)*log(3 - sqrt(6)) - (3/2 + 5*sqrt(6)/6)*log(3 + sqrt(6))
Respuesta numérica [src] 
0.0361728070050691
0.0361728070050691

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

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