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Resolución de integrales/ x^2-4/ (3x-1)/ (3x-1)/(4x^2-4x-7)

Integral de (3x-1)/(4x^2-4x-7) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                  
  /                  
 |                   
 |     3*x - 1       
 |  -------------- dx
 |     2             
 |  4*x  - 4*x - 7   
 |                   
/                    
0
∫013x−1(4x2−4x)−7 dx\int\limits_{0}^{1} \frac{3 x - 1}{\left(4 x^{2} - 4 x\right) - 7}\, dx 
Integral((3*x - 1)/(4*x^2 - 4*x - 7), (x, 0, 1))
Respuesta (Indefinida) [src] 
                           /            /  ___           \                                               
                           |   ___      |\/ 2 *(-1/2 + x)|                                               
                           |-\/ 2 *acoth|----------------|                                               
                           |            \       2        /                 2                             
                           |-------------------------------  for (-1/2 + x)  > 2                         
                           |               2                                                             
                           <                                                                             
                           |            /  ___           \                                               
                           |   ___      |\/ 2 *(-1/2 + x)|                                               
                           |-\/ 2 *atanh|----------------|                                               
  /                        |            \       2        /                 2                             
 |                         |-------------------------------  for (-1/2 + x)  < 2        /              2\
 |    3*x - 1              \               2                                       3*log\-7 - 4*x + 4*x /
 | -------------- dx = C + ----------------------------------------------------- + ----------------------
 |    2                                              8                                       8           
 | 4*x  - 4*x - 7                                                                                        
 |                                                                                                       
/
∫3x−1(4x2−4x)−7 dx=C+{−2acoth⁡(2(x−12)2)2for (x−12)2>2−2atanh⁡(2(x−12)2)2for (x−12)2<28+3log⁡(4x2−4x−7)8\int \frac{3 x - 1}{\left(4 x^{2} - 4 x\right) - 7}\, dx = C + \frac{\begin{cases} - \frac{\sqrt{2} \operatorname{acoth}{\left(\frac{\sqrt{2} \left(x - \frac{1}{2}\right)}{2} \right)}}{2} & \text{for}\: \left(x - \frac{1}{2}\right)^{2} > 2 \\- \frac{\sqrt{2} \operatorname{atanh}{\left(\frac{\sqrt{2} \left(x - \frac{1}{2}\right)}{2} \right)}}{2} & \text{for}\: \left(x - \frac{1}{2}\right)^{2} < 2 \end{cases}}{8} + \frac{3 \log{\left(4 x^{2} - 4 x - 7 \right)}}{8}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.900.5-0.5
Trazar el gráfico f(x)
Respuesta [src] 
/      ___\                  /      ___\                             /      ___\                    /      ___\                        
|3   \/ 2 |    /1     ___\   |3   \/ 2 | /          /  1     ___\\   |3   \/ 2 |    /  1     ___\   |3   \/ 2 | /          /1     ___\\
|- - -----|*log|- + \/ 2 | + |- + -----|*|pi*I + log|- - + \/ 2 || - |- - -----|*log|- - + \/ 2 | - |- + -----|*|pi*I + log|- + \/ 2 ||
\8     32 /    \2        /   \8     32 / \          \  2        //   \8     32 /    \  2        /   \8     32 / \          \2        //
−(38−232)log⁡(−12+2)+(38−232)log⁡(12+2)−(232+38)(log⁡(12+2)+iπ)+(232+38)(log⁡(−12+2)+iπ)- \left(\frac{3}{8} - \frac{\sqrt{2}}{32}\right) \log{\left(- \frac{1}{2} + \sqrt{2} \right)} + \left(\frac{3}{8} - \frac{\sqrt{2}}{32}\right) \log{\left(\frac{1}{2} + \sqrt{2} \right)} - \left(\frac{\sqrt{2}}{32} + \frac{3}{8}\right) \left(\log{\left(\frac{1}{2} + \sqrt{2} \right)} + i \pi\right) + \left(\frac{\sqrt{2}}{32} + \frac{3}{8}\right) \left(\log{\left(- \frac{1}{2} + \sqrt{2} \right)} + i \pi\right) 
=
= 
/      ___\                  /      ___\                             /      ___\                    /      ___\                        
|3   \/ 2 |    /1     ___\   |3   \/ 2 | /          /  1     ___\\   |3   \/ 2 |    /  1     ___\   |3   \/ 2 | /          /1     ___\\
|- - -----|*log|- + \/ 2 | + |- + -----|*|pi*I + log|- - + \/ 2 || - |- - -----|*log|- - + \/ 2 | - |- + -----|*|pi*I + log|- + \/ 2 ||
\8     32 /    \2        /   \8     32 / \          \  2        //   \8     32 /    \  2        /   \8     32 / \          \2        //
−(38−232)log⁡(−12+2)+(38−232)log⁡(12+2)−(232+38)(log⁡(12+2)+iπ)+(232+38)(log⁡(−12+2)+iπ)- \left(\frac{3}{8} - \frac{\sqrt{2}}{32}\right) \log{\left(- \frac{1}{2} + \sqrt{2} \right)} + \left(\frac{3}{8} - \frac{\sqrt{2}}{32}\right) \log{\left(\frac{1}{2} + \sqrt{2} \right)} - \left(\frac{\sqrt{2}}{32} + \frac{3}{8}\right) \left(\log{\left(\frac{1}{2} + \sqrt{2} \right)} + i \pi\right) + \left(\frac{\sqrt{2}}{32} + \frac{3}{8}\right) \left(\log{\left(- \frac{1}{2} + \sqrt{2} \right)} + i \pi\right) 
(3/8 - sqrt(2)/32)*log(1/2 + sqrt(2)) + (3/8 + sqrt(2)/32)*(pi*i + log(-1/2 + sqrt(2))) - (3/8 - sqrt(2)/32)*log(-1/2 + sqrt(2)) - (3/8 + sqrt(2)/32)*(pi*i + log(1/2 + sqrt(2)))
Respuesta numérica [src] 
-0.06531880717256
-0.06531880717256

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

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