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Resolución de integrales/ (3-x)/ x-x^2/ (3-x)/(√16+16*x-x^2)

Integral de (3-x)/(√16+16*x-x^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                      
  /                      
 |                       
 |        3 - x          
 |  ------------------ dx
 |    ____           2   
 |  \/ 16  + 16*x - x    
 |                       
/                        
0
$$\int\limits_{0}^{1} \frac{3 - x}{- x^{2} + \left(16 x + \sqrt{16}\right)}\, dx$$ 
Integral((3 - x)/(sqrt(16) + 16*x - x^2), (x, 0, 1))
Respuesta (Indefinida) [src] 
                                                           //             /  ____         \                     \
                                                           ||   ____      |\/ 17 *(-8 + x)|                     |
                                                           ||-\/ 17 *acoth|---------------|                     |
  /                                                        ||             \       34      /               2     |
 |                                /  ____    2       \     ||-------------------------------  for (-8 + x)  > 68|
 |       3 - x                 log\\/ 16  - x  + 16*x/     ||               34                                  |
 | ------------------ dx = C + ----------------------- + 5*|<                                                   |
 |   ____           2                     2                ||             /  ____         \                     |
 | \/ 16  + 16*x - x                                       ||   ____      |\/ 17 *(-8 + x)|                     |
 |                                                         ||-\/ 17 *atanh|---------------|                     |
/                                                          ||             \       34      /               2     |
                                                           ||-------------------------------  for (-8 + x)  < 68|
                                                           \\               34                                  /
$$\int \frac{3 - x}{- x^{2} + \left(16 x + \sqrt{16}\right)}\, dx = C + 5 \left(\begin{cases} - \frac{\sqrt{17} \operatorname{acoth}{\left(\frac{\sqrt{17} \left(x - 8\right)}{34} \right)}}{34} & \text{for}\: \left(x - 8\right)^{2} > 68 \\- \frac{\sqrt{17} \operatorname{atanh}{\left(\frac{\sqrt{17} \left(x - 8\right)}{34} \right)}}{34} & \text{for}\: \left(x - 8\right)^{2} < 68 \end{cases}\right) + \frac{\log{\left(- x^{2} + 16 x + \sqrt{16} \right)}}{2}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
/        ____\                      /        ____\                              /        ____\                      /        ____\                           
|1   5*\/ 17 |    /         ____\   |1   5*\/ 17 | /          /        ____\\   |1   5*\/ 17 |    /         ____\   |1   5*\/ 17 | /          /        ____\\
|- - --------|*log\-7 + 2*\/ 17 / + |- + --------|*\pi*I + log\7 + 2*\/ 17 // - |- - --------|*log\-8 + 2*\/ 17 / - |- + --------|*\pi*I + log\8 + 2*\/ 17 //
\2      68   /                      \2      68   /                              \2      68   /                      \2      68   /
$$\left(\frac{1}{2} - \frac{5 \sqrt{17}}{68}\right) \log{\left(-7 + 2 \sqrt{17} \right)} - \left(\frac{1}{2} - \frac{5 \sqrt{17}}{68}\right) \log{\left(-8 + 2 \sqrt{17} \right)} - \left(\frac{5 \sqrt{17}}{68} + \frac{1}{2}\right) \left(\log{\left(8 + 2 \sqrt{17} \right)} + i \pi\right) + \left(\frac{5 \sqrt{17}}{68} + \frac{1}{2}\right) \left(\log{\left(7 + 2 \sqrt{17} \right)} + i \pi\right)$$ 
=
= 
/        ____\                      /        ____\                              /        ____\                      /        ____\                           
|1   5*\/ 17 |    /         ____\   |1   5*\/ 17 | /          /        ____\\   |1   5*\/ 17 |    /         ____\   |1   5*\/ 17 | /          /        ____\\
|- - --------|*log\-7 + 2*\/ 17 / + |- + --------|*\pi*I + log\7 + 2*\/ 17 // - |- - --------|*log\-8 + 2*\/ 17 / - |- + --------|*\pi*I + log\8 + 2*\/ 17 //
\2      68   /                      \2      68   /                              \2      68   /                      \2      68   /
$$\left(\frac{1}{2} - \frac{5 \sqrt{17}}{68}\right) \log{\left(-7 + 2 \sqrt{17} \right)} - \left(\frac{1}{2} - \frac{5 \sqrt{17}}{68}\right) \log{\left(-8 + 2 \sqrt{17} \right)} - \left(\frac{5 \sqrt{17}}{68} + \frac{1}{2}\right) \left(\log{\left(8 + 2 \sqrt{17} \right)} + i \pi\right) + \left(\frac{5 \sqrt{17}}{68} + \frac{1}{2}\right) \left(\log{\left(7 + 2 \sqrt{17} \right)} + i \pi\right)$$ 
(1/2 - 5*sqrt(17)/68)*log(-7 + 2*sqrt(17)) + (1/2 + 5*sqrt(17)/68)*(pi*i + log(7 + 2*sqrt(17))) - (1/2 - 5*sqrt(17)/68)*log(-8 + 2*sqrt(17)) - (1/2 + 5*sqrt(17)/68)*(pi*i + log(8 + 2*sqrt(17)))
Respuesta numérica [src] 
0.268170404179652
0.268170404179652

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

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