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Resolución de integrales/ x^2-2/ x^2+5/ (1/(x^2-25))+(1/√(x^2+5))

Integral de (1/(x^2-25))+(1/√(x^2+5)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                           
  /                           
 |                            
 |  /   1           1     \   
 |  |------- + -----------| dx
 |  | 2           ________|   
 |  |x  - 25     /  2     |   
 |  \          \/  x  + 5 /   
 |                            
/                             
0
∫01(1x2+5+1x2−25) dx\int\limits_{0}^{1} \left(\frac{1}{\sqrt{x^{2} + 5}} + \frac{1}{x^{2} - 25}\right)\, dx 
Integral(1/(x^2 - 25) + 1/(sqrt(x^2 + 5)), (x, 0, 1))
Solución detallada

  1. Integramos término a término:

      TrigSubstitutionRule(theta=_theta, func=sqrt(5)*tan(_theta), rewritten=sec(_theta), substep=RewriteRule(rewritten=(tan(_theta)*sec(_theta) + sec(_theta)**2)/(tan(_theta) + sec(_theta)), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=tan(_theta) + sec(_theta), constant=1, substep=ReciprocalRule(func=_u, context=1/_u, symbol=_u), context=(tan(_theta)*sec(_theta) + sec(_theta)**2)/(tan(_theta) + sec(_theta)), symbol=_theta)], context=(tan(_theta)*sec(_theta) + sec(_theta)**2)/(tan(_theta) + sec(_theta)), symbol=_theta), context=sec(_theta), symbol=_theta), restriction=True, context=1/(sqrt(x**2 + 5)), symbol=x)

      PieceweseRule(subfunctions=[(ArctanRule(a=1, b=1, c=-25, context=1/(x**2 - 25), symbol=x), False), (ArccothRule(a=1, b=1, c=-25, context=1/(x**2 - 25), symbol=x), x**2 > 25), (ArctanhRule(a=1, b=1, c=-25, context=1/(x**2 - 25), symbol=x), x**2 < 25)], context=1/(x**2 - 25), symbol=x)

    El resultado es: {−acoth⁡(x5)5for x2>25−atanh⁡(x5)5for x2<25+log⁡(5x5+x25+1)\begin{cases} - \frac{\operatorname{acoth}{\left(\frac{x}{5} \right)}}{5} & \text{for}\: x^{2} > 25 \\- \frac{\operatorname{atanh}{\left(\frac{x}{5} \right)}}{5} & \text{for}\: x^{2} < 25 \end{cases} + \log{\left(\frac{\sqrt{5} x}{5} + \sqrt{\frac{x^{2}}{5} + 1} \right)}

  2. Ahora simplificar:

    {log⁡(x+x2+5)−acoth⁡(x5)5−log⁡(5)2for x2>25log⁡(x+x2+5)−atanh⁡(x5)5−log⁡(5)2for x2<25\begin{cases} \log{\left(x + \sqrt{x^{2} + 5} \right)} - \frac{\operatorname{acoth}{\left(\frac{x}{5} \right)}}{5} - \frac{\log{\left(5 \right)}}{2} & \text{for}\: x^{2} > 25 \\\log{\left(x + \sqrt{x^{2} + 5} \right)} - \frac{\operatorname{atanh}{\left(\frac{x}{5} \right)}}{5} - \frac{\log{\left(5 \right)}}{2} & \text{for}\: x^{2} < 25 \end{cases}

  3. Añadimos la constante de integración:

    {log⁡(x+x2+5)−acoth⁡(x5)5−log⁡(5)2for x2>25log⁡(x+x2+5)−atanh⁡(x5)5−log⁡(5)2for x2<25+constant\begin{cases} \log{\left(x + \sqrt{x^{2} + 5} \right)} - \frac{\operatorname{acoth}{\left(\frac{x}{5} \right)}}{5} - \frac{\log{\left(5 \right)}}{2} & \text{for}\: x^{2} > 25 \\\log{\left(x + \sqrt{x^{2} + 5} \right)} - \frac{\operatorname{atanh}{\left(\frac{x}{5} \right)}}{5} - \frac{\log{\left(5 \right)}}{2} & \text{for}\: x^{2} < 25 \end{cases}+ \mathrm{constant}

Respuesta:

{log⁡(x+x2+5)−acoth⁡(x5)5−log⁡(5)2for x2>25log⁡(x+x2+5)−atanh⁡(x5)5−log⁡(5)2for x2<25+constant\begin{cases} \log{\left(x + \sqrt{x^{2} + 5} \right)} - \frac{\operatorname{acoth}{\left(\frac{x}{5} \right)}}{5} - \frac{\log{\left(5 \right)}}{2} & \text{for}\: x^{2} > 25 \\\log{\left(x + \sqrt{x^{2} + 5} \right)} - \frac{\operatorname{atanh}{\left(\frac{x}{5} \right)}}{5} - \frac{\log{\left(5 \right)}}{2} & \text{for}\: x^{2} < 25 \end{cases}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                                    //      /x\              \                               
                                    ||-acoth|-|              |                               
  /                                 ||      \5/        2     |      /     ________          \
 |                                  ||----------  for x  > 25|      |    /      2        ___|
 | /   1           1     \          ||    5                  |      |   /      x     x*\/ 5 |
 | |------- + -----------| dx = C + |<                       | + log|  /   1 + --  + -------|
 | | 2           ________|          ||      /x\              |      \\/        5        5   /
 | |x  - 25     /  2     |          ||-atanh|-|              |                               
 | \          \/  x  + 5 /          ||      \5/        2     |                               
 |                                  ||----------  for x  < 25|                               
/                                   \\    5                  /
∫(1x2+5+1x2−25) dx=C+{−acoth⁡(x5)5for x2>25−atanh⁡(x5)5for x2<25+log⁡(5x5+x25+1)\int \left(\frac{1}{\sqrt{x^{2} + 5}} + \frac{1}{x^{2} - 25}\right)\, dx = C + \begin{cases} - \frac{\operatorname{acoth}{\left(\frac{x}{5} \right)}}{5} & \text{for}\: x^{2} > 25 \\- \frac{\operatorname{atanh}{\left(\frac{x}{5} \right)}}{5} & \text{for}\: x^{2} < 25 \end{cases} + \log{\left(\frac{\sqrt{5} x}{5} + \sqrt{\frac{x^{2}}{5} + 1} \right)}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.900.350.45
Trazar el gráfico f(x)
Respuesta [src] 
                         /  ___\
  log(6)   log(4)        |\/ 5 |
- ------ + ------ + asinh|-----|
    10       10          \  5  /
−log⁡(6)10+log⁡(4)10+asinh⁡(55)- \frac{\log{\left(6 \right)}}{10} + \frac{\log{\left(4 \right)}}{10} + \operatorname{asinh}{\left(\frac{\sqrt{5}}{5} \right)} 
=
= 
                         /  ___\
  log(6)   log(4)        |\/ 5 |
- ------ + ------ + asinh|-----|
    10       10          \  5  /
−log⁡(6)10+log⁡(4)10+asinh⁡(55)- \frac{\log{\left(6 \right)}}{10} + \frac{\log{\left(4 \right)}}{10} + \operatorname{asinh}{\left(\frac{\sqrt{5}}{5} \right)} 
-log(6)/10 + log(4)/10 + asinh(sqrt(5)/5)
Respuesta numérica [src] 
0.392960852434466
0.392960852434466

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

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