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Resolución de integrales/ 2x+12/ x^2+3/ (31+x)/(-x^2+32x+12)

Integral de (31+x)/(-x^2+32x+12) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                    
  /                    
 |                     
 |       31 + x        
 |  ---------------- dx
 |     2               
 |  - x  + 32*x + 12   
 |                     
/                      
0
$$\int\limits_{0}^{1} \frac{x + 31}{\left(- x^{2} + 32 x\right) + 12}\, dx$$ 
Integral((31 + x)/(-x^2 + 32*x + 12), (x, 0, 1))
Respuesta (Indefinida) [src] 
                                //             /  ____          \                       \                      
                                ||   ____      |\/ 67 *(-16 + x)|                       |                      
                                ||-\/ 67 *acoth|----------------|                       |                      
  /                             ||             \      134       /                2      |                      
 |                              ||--------------------------------  for (-16 + x)  > 268|      /      2       \
 |      31 + x                  ||              134                                     |   log\12 - x  + 32*x/
 | ---------------- dx = C - 47*|<                                                      | - -------------------
 |    2                         ||             /  ____          \                       |            2         
 | - x  + 32*x + 12             ||   ____      |\/ 67 *(-16 + x)|                       |                      
 |                              ||-\/ 67 *atanh|----------------|                       |                      
/                               ||             \      134       /                2      |                      
                                ||--------------------------------  for (-16 + x)  < 268|                      
                                \\              134                                     /
$$\int \frac{x + 31}{\left(- x^{2} + 32 x\right) + 12}\, dx = C - 47 \left(\begin{cases} - \frac{\sqrt{67} \operatorname{acoth}{\left(\frac{\sqrt{67} \left(x - 16\right)}{134} \right)}}{134} & \text{for}\: \left(x - 16\right)^{2} > 268 \\- \frac{\sqrt{67} \operatorname{atanh}{\left(\frac{\sqrt{67} \left(x - 16\right)}{134} \right)}}{134} & \text{for}\: \left(x - 16\right)^{2} < 268 \end{cases}\right) - \frac{\log{\left(- x^{2} + 32 x + 12 \right)}}{2}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
/         ____\                       /         ____\                               /         ____\                       /         ____\                            
|1   47*\/ 67 |    /          ____\   |1   47*\/ 67 | /          /         ____\\   |1   47*\/ 67 |    /          ____\   |1   47*\/ 67 | /          /         ____\\
|- - ---------|*log\-16 + 2*\/ 67 / + |- + ---------|*\pi*I + log\16 + 2*\/ 67 // - |- - ---------|*log\-15 + 2*\/ 67 / - |- + ---------|*\pi*I + log\15 + 2*\/ 67 //
\2      268   /                       \2      268   /                               \2      268   /                       \2      268   /
$$- \left(\frac{1}{2} - \frac{47 \sqrt{67}}{268}\right) \log{\left(-15 + 2 \sqrt{67} \right)} + \left(\frac{1}{2} - \frac{47 \sqrt{67}}{268}\right) \log{\left(-16 + 2 \sqrt{67} \right)} - \left(\frac{1}{2} + \frac{47 \sqrt{67}}{268}\right) \left(\log{\left(15 + 2 \sqrt{67} \right)} + i \pi\right) + \left(\frac{1}{2} + \frac{47 \sqrt{67}}{268}\right) \left(\log{\left(16 + 2 \sqrt{67} \right)} + i \pi\right)$$ 
=
= 
/         ____\                       /         ____\                               /         ____\                       /         ____\                            
|1   47*\/ 67 |    /          ____\   |1   47*\/ 67 | /          /         ____\\   |1   47*\/ 67 |    /          ____\   |1   47*\/ 67 | /          /         ____\\
|- - ---------|*log\-16 + 2*\/ 67 / + |- + ---------|*\pi*I + log\16 + 2*\/ 67 // - |- - ---------|*log\-15 + 2*\/ 67 / - |- + ---------|*\pi*I + log\15 + 2*\/ 67 //
\2      268   /                       \2      268   /                               \2      268   /                       \2      268   /
$$- \left(\frac{1}{2} - \frac{47 \sqrt{67}}{268}\right) \log{\left(-15 + 2 \sqrt{67} \right)} + \left(\frac{1}{2} - \frac{47 \sqrt{67}}{268}\right) \log{\left(-16 + 2 \sqrt{67} \right)} - \left(\frac{1}{2} + \frac{47 \sqrt{67}}{268}\right) \left(\log{\left(15 + 2 \sqrt{67} \right)} + i \pi\right) + \left(\frac{1}{2} + \frac{47 \sqrt{67}}{268}\right) \left(\log{\left(16 + 2 \sqrt{67} \right)} + i \pi\right)$$ 
(1/2 - 47*sqrt(67)/268)*log(-16 + 2*sqrt(67)) + (1/2 + 47*sqrt(67)/268)*(pi*i + log(16 + 2*sqrt(67))) - (1/2 - 47*sqrt(67)/268)*log(-15 + 2*sqrt(67)) - (1/2 + 47*sqrt(67)/268)*(pi*i + log(15 + 2*sqrt(67)))
Respuesta numérica [src] 
1.28405056385089
1.28405056385089

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

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