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Resolución de integrales/ x-x^2/ 1/((√5+2x-x^2))

Integral de 1/((√5+2x-x^2)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                    
  /                    
 |                     
 |         1           
 |  ---------------- dx
 |    ___          2   
 |  \/ 5  + 2*x - x    
 |                     
/                      
0
∫011−x2+(2x+5) dx\int\limits_{0}^{1} \frac{1}{- x^{2} + \left(2 x + \sqrt{5}\right)}\, dx 
Integral(1/(sqrt(5) + 2*x - x^2), (x, 0, 1))
Respuesta (Indefinida) [src] 
                             //      /    -1 + x    \                            \
                             ||-acoth|--------------|                            |
                             ||      |   ___________|                            |
                             ||      |  /       ___ |                            |
                             ||      \\/  1 + \/ 5  /               2         ___|
                             ||-----------------------  for (-1 + x)  > 1 + \/ 5 |
  /                          ||        ___________                               |
 |                           ||       /       ___                                |
 |        1                  ||     \/  1 + \/ 5                                 |
 | ---------------- dx = C - |<                                                  |
 |   ___          2          ||      /    -1 + x    \                            |
 | \/ 5  + 2*x - x           ||-atanh|--------------|                            |
 |                           ||      |   ___________|                            |
/                            ||      |  /       ___ |                            |
                             ||      \\/  1 + \/ 5  /               2         ___|
                             ||-----------------------  for (-1 + x)  < 1 + \/ 5 |
                             ||        ___________                               |
                             ||       /       ___                                |
                             \\     \/  1 + \/ 5                                 /
∫1−x2+(2x+5) dx=C−{−acoth⁡(x−11+5)1+5for (x−1)2>1+5−atanh⁡(x−11+5)1+5for (x−1)2<1+5\int \frac{1}{- x^{2} + \left(2 x + \sqrt{5}\right)}\, dx = C - \begin{cases} - \frac{\operatorname{acoth}{\left(\frac{x - 1}{\sqrt{1 + \sqrt{5}}} \right)}}{\sqrt{1 + \sqrt{5}}} & \text{for}\: \left(x - 1\right)^{2} > 1 + \sqrt{5} \\- \frac{\operatorname{atanh}{\left(\frac{x - 1}{\sqrt{1 + \sqrt{5}}} \right)}}{\sqrt{1 + \sqrt{5}}} & \text{for}\: \left(x - 1\right)^{2} < 1 + \sqrt{5} \end{cases}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.900.20.6
Trazar el gráfico f(x)
Respuesta [src] 
     ______________ /          /         ______________               \\        ______________    /      ______________               \        ______________ /          /      ______________               \\        ______________    /          ______________               \
    /          ___  |          |        /          ___                ||       /          ___     |     /          ___                |       /          ___  |          |     /          ___                ||       /          ___     |         /          ___                |
   /    1    \/ 5   |          |       /    1    \/ 5   /         ___\||      /    1    \/ 5      |    /    1    \/ 5   /         ___\|      /    1    \/ 5   |          |    /    1    \/ 5   /         ___\||      /    1    \/ 5      |        /    1    \/ 5   /         ___\|
  /   - -- + ----- *|pi*I + log|1 -   /   - -- + ----- *\-2 - 2*\/ 5 /|| +   /   - -- + ----- *log|-  /   - -- + ----- *\-2 - 2*\/ 5 /| -   /   - -- + ----- *|pi*I + log|-  /   - -- + ----- *\-2 - 2*\/ 5 /|| -   /   - -- + ----- *log|-1 -   /   - -- + ----- *\-2 - 2*\/ 5 /|
\/      16     16   \          \    \/      16     16                 //   \/      16     16      \ \/      16     16                 /   \/      16     16   \          \ \/      16     16                 //   \/      16     16      \     \/      16     16                 /
−−116+516log⁡(−1−−116+516(−25−2))+−116+516log⁡(−−116+516(−25−2))−−116+516(log⁡(−−116+516(−25−2))+iπ)+−116+516(log⁡(1−−116+516(−25−2))+iπ)- \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \log{\left(-1 - \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \left(- 2 \sqrt{5} - 2\right) \right)} + \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \log{\left(- \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \left(- 2 \sqrt{5} - 2\right) \right)} - \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \left(\log{\left(- \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \left(- 2 \sqrt{5} - 2\right) \right)} + i \pi\right) + \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \left(\log{\left(1 - \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \left(- 2 \sqrt{5} - 2\right) \right)} + i \pi\right) 
=
= 
     ______________ /          /         ______________               \\        ______________    /      ______________               \        ______________ /          /      ______________               \\        ______________    /          ______________               \
    /          ___  |          |        /          ___                ||       /          ___     |     /          ___                |       /          ___  |          |     /          ___                ||       /          ___     |         /          ___                |
   /    1    \/ 5   |          |       /    1    \/ 5   /         ___\||      /    1    \/ 5      |    /    1    \/ 5   /         ___\|      /    1    \/ 5   |          |    /    1    \/ 5   /         ___\||      /    1    \/ 5      |        /    1    \/ 5   /         ___\|
  /   - -- + ----- *|pi*I + log|1 -   /   - -- + ----- *\-2 - 2*\/ 5 /|| +   /   - -- + ----- *log|-  /   - -- + ----- *\-2 - 2*\/ 5 /| -   /   - -- + ----- *|pi*I + log|-  /   - -- + ----- *\-2 - 2*\/ 5 /|| -   /   - -- + ----- *log|-1 -   /   - -- + ----- *\-2 - 2*\/ 5 /|
\/      16     16   \          \    \/      16     16                 //   \/      16     16      \ \/      16     16                 /   \/      16     16   \          \ \/      16     16                 //   \/      16     16      \     \/      16     16                 /
−−116+516log⁡(−1−−116+516(−25−2))+−116+516log⁡(−−116+516(−25−2))−−116+516(log⁡(−−116+516(−25−2))+iπ)+−116+516(log⁡(1−−116+516(−25−2))+iπ)- \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \log{\left(-1 - \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \left(- 2 \sqrt{5} - 2\right) \right)} + \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \log{\left(- \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \left(- 2 \sqrt{5} - 2\right) \right)} - \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \left(\log{\left(- \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \left(- 2 \sqrt{5} - 2\right) \right)} + i \pi\right) + \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \left(\log{\left(1 - \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \left(- 2 \sqrt{5} - 2\right) \right)} + i \pi\right) 
sqrt(-1/16 + sqrt(5)/16)*(pi*i + log(1 - sqrt(-1/16 + sqrt(5)/16)*(-2 - 2*sqrt(5)))) + sqrt(-1/16 + sqrt(5)/16)*log(-sqrt(-1/16 + sqrt(5)/16)*(-2 - 2*sqrt(5))) - sqrt(-1/16 + sqrt(5)/16)*(pi*i + log(-sqrt(-1/16 + sqrt(5)/16)*(-2 - 2*sqrt(5)))) - sqrt(-1/16 + sqrt(5)/16)*log(-1 - sqrt(-1/16 + sqrt(5)/16)*(-2 - 2*sqrt(5)))
Respuesta numérica [src] 
0.348472438873422
0.348472438873422

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

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