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Resolución de integrales/ (4+2x)/ (4+2x)/(4-x-2x²)

Integral de (4+2x)/(4-x-2x²) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                
  /                
 |                 
 |    4 + 2*x      
 |  ------------ dx
 |             2   
 |  4 - x - 2*x    
 |                 
/                  
0
$$\int\limits_{0}^{1} \frac{2 x + 4}{- 2 x^{2} + \left(4 - x\right)}\, dx$$ 
Integral((4 + 2*x)/(4 - x - 2*x^2), (x, 0, 1))
Respuesta (Indefinida) [src] 
                            //             /    ____          \                      \                    
                            ||   ____      |4*\/ 33 *(1/4 + x)|                      |                    
                            ||-\/ 33 *acoth|------------------|                      |                    
  /                         ||             \        33        /                2   33|                    
 |                          ||----------------------------------  for (1/4 + x)  > --|      /           2\
 |   4 + 2*x                ||               132                                   16|   log\4 - x - 2*x /
 | ------------ dx = C - 28*|<                                                       | - -----------------
 |            2             ||             /    ____          \                      |           2        
 | 4 - x - 2*x              ||   ____      |4*\/ 33 *(1/4 + x)|                      |                    
 |                          ||-\/ 33 *atanh|------------------|                      |                    
/                           ||             \        33        /                2   33|                    
                            ||----------------------------------  for (1/4 + x)  < --|                    
                            \\               132                                   16/
$$\int \frac{2 x + 4}{- 2 x^{2} + \left(4 - x\right)}\, dx = C - 28 \left(\begin{cases} - \frac{\sqrt{33} \operatorname{acoth}{\left(\frac{4 \sqrt{33} \left(x + \frac{1}{4}\right)}{33} \right)}}{132} & \text{for}\: \left(x + \frac{1}{4}\right)^{2} > \frac{33}{16} \\- \frac{\sqrt{33} \operatorname{atanh}{\left(\frac{4 \sqrt{33} \left(x + \frac{1}{4}\right)}{33} \right)}}{132} & \text{for}\: \left(x + \frac{1}{4}\right)^{2} < \frac{33}{16} \end{cases}\right) - \frac{\log{\left(- 2 x^{2} - x + 4 \right)}}{2}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
/        ____\    /      ____\   /        ____\ /          /        ____\\   /        ____\    /      ____\   /        ____\ /          /        ____\\
|1   7*\/ 33 |    |1   \/ 33 |   |1   7*\/ 33 | |          |  1   \/ 33 ||   |1   7*\/ 33 |    |5   \/ 33 |   |1   7*\/ 33 | |          |  5   \/ 33 ||
|- - --------|*log|- + ------| + |- + --------|*|pi*I + log|- - + ------|| - |- - --------|*log|- + ------| - |- + --------|*|pi*I + log|- - + ------||
\2      66   /    \4     4   /   \2      66   / \          \  4     4   //   \2      66   /    \4     4   /   \2      66   / \          \  4     4   //
$$\left(\frac{1}{2} - \frac{7 \sqrt{33}}{66}\right) \log{\left(\frac{1}{4} + \frac{\sqrt{33}}{4} \right)} - \left(\frac{1}{2} - \frac{7 \sqrt{33}}{66}\right) \log{\left(\frac{5}{4} + \frac{\sqrt{33}}{4} \right)} - \left(\frac{1}{2} + \frac{7 \sqrt{33}}{66}\right) \left(\log{\left(- \frac{5}{4} + \frac{\sqrt{33}}{4} \right)} + i \pi\right) + \left(\frac{1}{2} + \frac{7 \sqrt{33}}{66}\right) \left(\log{\left(- \frac{1}{4} + \frac{\sqrt{33}}{4} \right)} + i \pi\right)$$ 
=
= 
/        ____\    /      ____\   /        ____\ /          /        ____\\   /        ____\    /      ____\   /        ____\ /          /        ____\\
|1   7*\/ 33 |    |1   \/ 33 |   |1   7*\/ 33 | |          |  1   \/ 33 ||   |1   7*\/ 33 |    |5   \/ 33 |   |1   7*\/ 33 | |          |  5   \/ 33 ||
|- - --------|*log|- + ------| + |- + --------|*|pi*I + log|- - + ------|| - |- - --------|*log|- + ------| - |- + --------|*|pi*I + log|- - + ------||
\2      66   /    \4     4   /   \2      66   / \          \  4     4   //   \2      66   /    \4     4   /   \2      66   / \          \  4     4   //
$$\left(\frac{1}{2} - \frac{7 \sqrt{33}}{66}\right) \log{\left(\frac{1}{4} + \frac{\sqrt{33}}{4} \right)} - \left(\frac{1}{2} - \frac{7 \sqrt{33}}{66}\right) \log{\left(\frac{5}{4} + \frac{\sqrt{33}}{4} \right)} - \left(\frac{1}{2} + \frac{7 \sqrt{33}}{66}\right) \left(\log{\left(- \frac{5}{4} + \frac{\sqrt{33}}{4} \right)} + i \pi\right) + \left(\frac{1}{2} + \frac{7 \sqrt{33}}{66}\right) \left(\log{\left(- \frac{1}{4} + \frac{\sqrt{33}}{4} \right)} + i \pi\right)$$ 
(1/2 - 7*sqrt(33)/66)*log(1/4 + sqrt(33)/4) + (1/2 + 7*sqrt(33)/66)*(pi*i + log(-1/4 + sqrt(33)/4)) - (1/2 - 7*sqrt(33)/66)*log(5/4 + sqrt(33)/4) - (1/2 + 7*sqrt(33)/66)*(pi*i + log(-5/4 + sqrt(33)/4))
Respuesta numérica [src] 
2.1052081192457
2.1052081192457

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

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