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Resolución de integrales/ sqrt(2)/ 1/(x*(sqrt(2)*(x))-9)

Integral de 1/(x*(sqrt(2)*(x))-9) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                 
  /                 
 |                  
 |        1         
 |  ------------- dx
 |      ___         
 |  x*\/ 2 *x - 9   
 |                  
/                   
0
∫011x2x−9 dx\int\limits_{0}^{1} \frac{1}{x \sqrt{2} x - 9}\, dx 
Integral(1/(x*(sqrt(2)*x) - 9), (x, 0, 1))
Solución detallada

    PieceweseRule(subfunctions=[(ArctanRule(a=1, b=sqrt(2), c=-9, context=1/(x*(sqrt(2)*x) - 9), symbol=x), False), (ArccothRule(a=1, b=sqrt(2), c=-9, context=1/(x*(sqrt(2)*x) - 9), symbol=x), x**2 > 9*sqrt(2)/2), (ArctanhRule(a=1, b=sqrt(2), c=-9, context=1/(x*(sqrt(2)*x) - 9), symbol=x), x**2 < 9*sqrt(2)/2)], context=1/(x*(sqrt(2)*x) - 9), symbol=x)

  1. Añadimos la constante de integración:

    {−234acoth⁡(24x3)6for x2>922−234atanh⁡(24x3)6for x2<922+constant\begin{cases} - \frac{2^{\frac{3}{4}} \operatorname{acoth}{\left(\frac{\sqrt[4]{2} x}{3} \right)}}{6} & \text{for}\: x^{2} > \frac{9 \sqrt{2}}{2} \\- \frac{2^{\frac{3}{4}} \operatorname{atanh}{\left(\frac{\sqrt[4]{2} x}{3} \right)}}{6} & \text{for}\: x^{2} < \frac{9 \sqrt{2}}{2} \end{cases}+ \mathrm{constant}

Respuesta:

{−234acoth⁡(24x3)6for x2>922−234atanh⁡(24x3)6for x2<922+constant\begin{cases} - \frac{2^{\frac{3}{4}} \operatorname{acoth}{\left(\frac{\sqrt[4]{2} x}{3} \right)}}{6} & \text{for}\: x^{2} > \frac{9 \sqrt{2}}{2} \\- \frac{2^{\frac{3}{4}} \operatorname{atanh}{\left(\frac{\sqrt[4]{2} x}{3} \right)}}{6} & \text{for}\: x^{2} < \frac{9 \sqrt{2}}{2} \end{cases}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                          //           /  4 ___\                   \
                          ||  3/4      |x*\/ 2 |                   |
                          ||-2   *acoth|-------|                ___|
  /                       ||           \   3   /        2   9*\/ 2 |
 |                        ||---------------------  for x  > -------|
 |       1                ||          6                        2   |
 | ------------- dx = C + |<                                       |
 |     ___                ||           /  4 ___\                   |
 | x*\/ 2 *x - 9          ||  3/4      |x*\/ 2 |                   |
 |                        ||-2   *atanh|-------|                ___|
/                         ||           \   3   /        2   9*\/ 2 |
                          ||---------------------  for x  < -------|
                          \\          6                        2   /
∫1x2x−9 dx=C+{−234acoth⁡(24x3)6for x2>922−234atanh⁡(24x3)6for x2<922\int \frac{1}{x \sqrt{2} x - 9}\, dx = C + \begin{cases} - \frac{2^{\frac{3}{4}} \operatorname{acoth}{\left(\frac{\sqrt[4]{2} x}{3} \right)}}{6} & \text{for}\: x^{2} > \frac{9 \sqrt{2}}{2} \\- \frac{2^{\frac{3}{4}} \operatorname{atanh}{\left(\frac{\sqrt[4]{2} x}{3} \right)}}{6} & \text{for}\: x^{2} < \frac{9 \sqrt{2}}{2} \end{cases}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.90-0.150-0.100
Trazar el gráfico f(x)
Respuesta [src] 
       /          /   3/4\\           /       3/4\        /          /        3/4\\           /   3/4\
   3/4 |          |3*2   ||    3/4    |    3*2   |    3/4 |          |     3*2   ||    3/4    |3*2   |
  2   *|pi*I + log|------||   2   *log|1 + ------|   2   *|pi*I + log|-1 + ------||   2   *log|------|
       \          \  2   //           \      2   /        \          \       2   //           \  2   /
- ------------------------- - -------------------- + ------------------------------ + ----------------
              12                       12                          12                        12
−234log⁡(1+3⋅2342)12+234log⁡(3⋅2342)12−234(log⁡(3⋅2342)+iπ)12+234(log⁡(−1+3⋅2342)+iπ)12- \frac{2^{\frac{3}{4}} \log{\left(1 + \frac{3 \cdot 2^{\frac{3}{4}}}{2} \right)}}{12} + \frac{2^{\frac{3}{4}} \log{\left(\frac{3 \cdot 2^{\frac{3}{4}}}{2} \right)}}{12} - \frac{2^{\frac{3}{4}} \left(\log{\left(\frac{3 \cdot 2^{\frac{3}{4}}}{2} \right)} + i \pi\right)}{12} + \frac{2^{\frac{3}{4}} \left(\log{\left(-1 + \frac{3 \cdot 2^{\frac{3}{4}}}{2} \right)} + i \pi\right)}{12} 
=
= 
       /          /   3/4\\           /       3/4\        /          /        3/4\\           /   3/4\
   3/4 |          |3*2   ||    3/4    |    3*2   |    3/4 |          |     3*2   ||    3/4    |3*2   |
  2   *|pi*I + log|------||   2   *log|1 + ------|   2   *|pi*I + log|-1 + ------||   2   *log|------|
       \          \  2   //           \      2   /        \          \       2   //           \  2   /
- ------------------------- - -------------------- + ------------------------------ + ----------------
              12                       12                          12                        12
−234log⁡(1+3⋅2342)12+234log⁡(3⋅2342)12−234(log⁡(3⋅2342)+iπ)12+234(log⁡(−1+3⋅2342)+iπ)12- \frac{2^{\frac{3}{4}} \log{\left(1 + \frac{3 \cdot 2^{\frac{3}{4}}}{2} \right)}}{12} + \frac{2^{\frac{3}{4}} \log{\left(\frac{3 \cdot 2^{\frac{3}{4}}}{2} \right)}}{12} - \frac{2^{\frac{3}{4}} \left(\log{\left(\frac{3 \cdot 2^{\frac{3}{4}}}{2} \right)} + i \pi\right)}{12} + \frac{2^{\frac{3}{4}} \left(\log{\left(-1 + \frac{3 \cdot 2^{\frac{3}{4}}}{2} \right)} + i \pi\right)}{12} 
-2^(3/4)*(pi*i + log(3*2^(3/4)/2))/12 - 2^(3/4)*log(1 + 3*2^(3/4)/2)/12 + 2^(3/4)*(pi*i + log(-1 + 3*2^(3/4)/2))/12 + 2^(3/4)*log(3*2^(3/4)/2)/12
Respuesta numérica [src] 
-0.117549845520564
-0.117549845520564

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

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