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Resolución de integrales/ (x-5)/ sqrt(x-5)/x^3

Integral de sqrt(x-5)/x^3 dx

Solución

Ha introducido [src]

  1             
  /             
 |              
 |    _______   
 |  \/ x - 5    
 |  --------- dx
 |       3      
 |      x       
 |              
/               
0

$$\int\limits_{0}^{1} \frac{\sqrt{x - 5}}{x^{3}}\, dx$$

Integral(sqrt(x - 5)/x^3, (x, 0, 1))

Solución detallada

que $u = \sqrt{x - 5}$ .

Luego que $du = \frac{dx}{2 \sqrt{x - 5}}$ y ponemos $2 du$ :

$\int \frac{2 u^{2}}{\left(u^{2} + 5\right)^{3}}\, du$
1. La integral del producto de una función por una constante es la constante por la integral de esta función:
  
  $\int \frac{u^{2}}{\left(u^{2} + 5\right)^{3}}\, du = 2 \int \frac{u^{2}}{\left(u^{2} + 5\right)^{3}}\, du$
  1. Hay varias maneras de calcular esta integral.
    Método #1
    1. Vuelva a escribir el integrando:
      
      $\frac{u^{2}}{\left(u^{2} + 5\right)^{3}} = \frac{1}{\left(u^{2} + 5\right)^{2}} - \frac{5}{\left(u^{2} + 5\right)^{3}}$
    2. Integramos término a término:
      
      TrigSubstitutionRule(theta=_theta, func=sqrt(5)*tan(_theta), rewritten=sqrt(5)*cos(_theta)**2/25, substep=ConstantTimesRule(constant=sqrt(5)/25, other=cos(_theta)**2, substep=RewriteRule(rewritten=cos(2*_theta)/2 + 1/2, substep=AddRule(substeps=[ConstantTimesRule(constant=1/2, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=cos(2*_theta)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(2*_theta)/2 + 1/2, symbol=_theta), context=cos(_theta)**2, symbol=_theta), context=sqrt(5)*cos(_theta)**2/25, symbol=_theta), restriction=True, context=(_u**2 + 5)**(-2), symbol=_u)
      
      La integral del producto de una función por una constante es la constante por la integral de esta función:
      
      $\int \left(- \frac{5}{\left(u^{2} + 5\right)^{3}}\right)\, du = - 5 \int \frac{1}{\left(u^{2} + 5\right)^{3}}\, du$
      
      TrigSubstitutionRule(theta=_theta, func=sqrt(5)*tan(_theta), rewritten=sqrt(5)*cos(_theta)**4/125, substep=ConstantTimesRule(constant=sqrt(5)/125, other=cos(_theta)**4, substep=RewriteRule(rewritten=(cos(2*_theta)/2 + 1/2)**2, substep=AlternativeRule(alternatives=[RewriteRule(rewritten=cos(2*_theta)**2/4 + cos(2*_theta)/2 + 1/4, substep=AddRule(substeps=[ConstantTimesRule(constant=1/4, other=cos(2*_theta)**2, substep=RewriteRule(rewritten=cos(4*_theta)/2 + 1/2, substep=AddRule(substeps=[ConstantTimesRule(constant=1/2, other=cos(4*_theta), substep=URule(u_var=_u, u_func=4*_theta, constant=1/4, substep=ConstantTimesRule(constant=1/4, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(4*_theta), symbol=_theta), context=cos(4*_theta)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(4*_theta)/2 + 1/2, symbol=_theta), context=cos(2*_theta)**2, symbol=_theta), context=cos(2*_theta)**2/4, symbol=_theta), ConstantTimesRule(constant=1/2, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=cos(2*_theta)/2, symbol=_theta), ConstantRule(constant=1/4, context=1/4, symbol=_theta)], context=cos(2*_theta)**2/4 + cos(2*_theta)/2 + 1/4, symbol=_theta), context=(cos(2*_theta)/2 + 1/2)**2, symbol=_theta), RewriteRule(rewritten=cos(2*_theta)**2/4 + cos(2*_theta)/2 + 1/4, substep=AddRule(substeps=[ConstantTimesRule(constant=1/4, other=cos(2*_theta)**2, substep=RewriteRule(rewritten=cos(4*_theta)/2 + 1/2, substep=AddRule(substeps=[ConstantTimesRule(constant=1/2, other=cos(4*_theta), substep=URule(u_var=_u, u_func=4*_theta, constant=1/4, substep=ConstantTimesRule(constant=1/4, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(4*_theta), symbol=_theta), context=cos(4*_theta)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(4*_theta)/2 + 1/2, symbol=_theta), context=cos(2*_theta)**2, symbol=_theta), context=cos(2*_theta)**2/4, symbol=_theta), ConstantTimesRule(constant=1/2, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=cos(2*_theta)/2, symbol=_theta), ConstantRule(constant=1/4, context=1/4, symbol=_theta)], context=cos(2*_theta)**2/4 + cos(2*_theta)/2 + 1/4, symbol=_theta), context=(cos(2*_theta)/2 + 1/2)**2, symbol=_theta)], context=(cos(2*_theta)/2 + 1/2)**2, symbol=_theta), context=cos(_theta)**4, symbol=_theta), context=sqrt(5)*cos(_theta)**4/125, symbol=_theta), restriction=True, context=(_u**2 + 5)**(-3), symbol=_u)
      
      Por lo tanto, el resultado es: $- \frac{\sqrt{5} \left(\frac{\sqrt{5} u \left(5 - u^{2}\right)}{8 \left(u^{2} + 5\right)^{2}} + \frac{\sqrt{5} u}{2 \left(u^{2} + 5\right)} + \frac{3 \operatorname{atan}{\left(\frac{\sqrt{5} u}{5} \right)}}{8}\right)}{25}$
      
      El resultado es: $\frac{\sqrt{5} \left(\frac{\sqrt{5} u}{2 \left(u^{2} + 5\right)} + \frac{\operatorname{atan}{\left(\frac{\sqrt{5} u}{5} \right)}}{2}\right)}{25} - \frac{\sqrt{5} \left(\frac{\sqrt{5} u \left(5 - u^{2}\right)}{8 \left(u^{2} + 5\right)^{2}} + \frac{\sqrt{5} u}{2 \left(u^{2} + 5\right)} + \frac{3 \operatorname{atan}{\left(\frac{\sqrt{5} u}{5} \right)}}{8}\right)}{25}$
    Método #2
    1. Vuelva a escribir el integrando:
      
      $\frac{u^{2}}{\left(u^{2} + 5\right)^{3}} = \frac{u^{2}}{u^{6} + 15 u^{4} + 75 u^{2} + 125}$
    2. Vuelva a escribir el integrando:
      
      $\frac{u^{2}}{u^{6} + 15 u^{4} + 75 u^{2} + 125} = \frac{1}{\left(u^{2} + 5\right)^{2}} - \frac{5}{\left(u^{2} + 5\right)^{3}}$
    3. Integramos término a término:
      
      TrigSubstitutionRule(theta=_theta, func=sqrt(5)*tan(_theta), rewritten=sqrt(5)*cos(_theta)**2/25, substep=ConstantTimesRule(constant=sqrt(5)/25, other=cos(_theta)**2, substep=RewriteRule(rewritten=cos(2*_theta)/2 + 1/2, substep=AddRule(substeps=[ConstantTimesRule(constant=1/2, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=cos(2*_theta)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(2*_theta)/2 + 1/2, symbol=_theta), context=cos(_theta)**2, symbol=_theta), context=sqrt(5)*cos(_theta)**2/25, symbol=_theta), restriction=True, context=(_u**2 + 5)**(-2), symbol=_u)
      
      La integral del producto de una función por una constante es la constante por la integral de esta función:
      
      $\int \left(- \frac{5}{\left(u^{2} + 5\right)^{3}}\right)\, du = - 5 \int \frac{1}{\left(u^{2} + 5\right)^{3}}\, du$
      
      TrigSubstitutionRule(theta=_theta, func=sqrt(5)*tan(_theta), rewritten=sqrt(5)*cos(_theta)**4/125, substep=ConstantTimesRule(constant=sqrt(5)/125, other=cos(_theta)**4, substep=RewriteRule(rewritten=(cos(2*_theta)/2 + 1/2)**2, substep=AlternativeRule(alternatives=[RewriteRule(rewritten=cos(2*_theta)**2/4 + cos(2*_theta)/2 + 1/4, substep=AddRule(substeps=[ConstantTimesRule(constant=1/4, other=cos(2*_theta)**2, substep=RewriteRule(rewritten=cos(4*_theta)/2 + 1/2, substep=AddRule(substeps=[ConstantTimesRule(constant=1/2, other=cos(4*_theta), substep=URule(u_var=_u, u_func=4*_theta, constant=1/4, substep=ConstantTimesRule(constant=1/4, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(4*_theta), symbol=_theta), context=cos(4*_theta)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(4*_theta)/2 + 1/2, symbol=_theta), context=cos(2*_theta)**2, symbol=_theta), context=cos(2*_theta)**2/4, symbol=_theta), ConstantTimesRule(constant=1/2, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=cos(2*_theta)/2, symbol=_theta), ConstantRule(constant=1/4, context=1/4, symbol=_theta)], context=cos(2*_theta)**2/4 + cos(2*_theta)/2 + 1/4, symbol=_theta), context=(cos(2*_theta)/2 + 1/2)**2, symbol=_theta), RewriteRule(rewritten=cos(2*_theta)**2/4 + cos(2*_theta)/2 + 1/4, substep=AddRule(substeps=[ConstantTimesRule(constant=1/4, other=cos(2*_theta)**2, substep=RewriteRule(rewritten=cos(4*_theta)/2 + 1/2, substep=AddRule(substeps=[ConstantTimesRule(constant=1/2, other=cos(4*_theta), substep=URule(u_var=_u, u_func=4*_theta, constant=1/4, substep=ConstantTimesRule(constant=1/4, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(4*_theta), symbol=_theta), context=cos(4*_theta)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(4*_theta)/2 + 1/2, symbol=_theta), context=cos(2*_theta)**2, symbol=_theta), context=cos(2*_theta)**2/4, symbol=_theta), ConstantTimesRule(constant=1/2, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=cos(2*_theta)/2, symbol=_theta), ConstantRule(constant=1/4, context=1/4, symbol=_theta)], context=cos(2*_theta)**2/4 + cos(2*_theta)/2 + 1/4, symbol=_theta), context=(cos(2*_theta)/2 + 1/2)**2, symbol=_theta)], context=(cos(2*_theta)/2 + 1/2)**2, symbol=_theta), context=cos(_theta)**4, symbol=_theta), context=sqrt(5)*cos(_theta)**4/125, symbol=_theta), restriction=True, context=(_u**2 + 5)**(-3), symbol=_u)
      
      Por lo tanto, el resultado es: $- \frac{\sqrt{5} \left(\frac{\sqrt{5} u \left(5 - u^{2}\right)}{8 \left(u^{2} + 5\right)^{2}} + \frac{\sqrt{5} u}{2 \left(u^{2} + 5\right)} + \frac{3 \operatorname{atan}{\left(\frac{\sqrt{5} u}{5} \right)}}{8}\right)}{25}$
      
      El resultado es: $\frac{\sqrt{5} \left(\frac{\sqrt{5} u}{2 \left(u^{2} + 5\right)} + \frac{\operatorname{atan}{\left(\frac{\sqrt{5} u}{5} \right)}}{2}\right)}{25} - \frac{\sqrt{5} \left(\frac{\sqrt{5} u \left(5 - u^{2}\right)}{8 \left(u^{2} + 5\right)^{2}} + \frac{\sqrt{5} u}{2 \left(u^{2} + 5\right)} + \frac{3 \operatorname{atan}{\left(\frac{\sqrt{5} u}{5} \right)}}{8}\right)}{25}$
    Método #3
    1. Vuelva a escribir el integrando:
      
      $\frac{u^{2}}{\left(u^{2} + 5\right)^{3}} = \frac{u^{2}}{u^{6} + 15 u^{4} + 75 u^{2} + 125}$
    2. Vuelva a escribir el integrando:
      
      $\frac{u^{2}}{u^{6} + 15 u^{4} + 75 u^{2} + 125} = \frac{1}{\left(u^{2} + 5\right)^{2}} - \frac{5}{\left(u^{2} + 5\right)^{3}}$
    3. Integramos término a término:
      
      TrigSubstitutionRule(theta=_theta, func=sqrt(5)*tan(_theta), rewritten=sqrt(5)*cos(_theta)**2/25, substep=ConstantTimesRule(constant=sqrt(5)/25, other=cos(_theta)**2, substep=RewriteRule(rewritten=cos(2*_theta)/2 + 1/2, substep=AddRule(substeps=[ConstantTimesRule(constant=1/2, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=cos(2*_theta)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(2*_theta)/2 + 1/2, symbol=_theta), context=cos(_theta)**2, symbol=_theta), context=sqrt(5)*cos(_theta)**2/25, symbol=_theta), restriction=True, context=(_u**2 + 5)**(-2), symbol=_u)
      
      La integral del producto de una función por una constante es la constante por la integral de esta función:
      
      $\int \left(- \frac{5}{\left(u^{2} + 5\right)^{3}}\right)\, du = - 5 \int \frac{1}{\left(u^{2} + 5\right)^{3}}\, du$
      
      TrigSubstitutionRule(theta=_theta, func=sqrt(5)*tan(_theta), rewritten=sqrt(5)*cos(_theta)**4/125, substep=ConstantTimesRule(constant=sqrt(5)/125, other=cos(_theta)**4, substep=RewriteRule(rewritten=(cos(2*_theta)/2 + 1/2)**2, substep=AlternativeRule(alternatives=[RewriteRule(rewritten=cos(2*_theta)**2/4 + cos(2*_theta)/2 + 1/4, substep=AddRule(substeps=[ConstantTimesRule(constant=1/4, other=cos(2*_theta)**2, substep=RewriteRule(rewritten=cos(4*_theta)/2 + 1/2, substep=AddRule(substeps=[ConstantTimesRule(constant=1/2, other=cos(4*_theta), substep=URule(u_var=_u, u_func=4*_theta, constant=1/4, substep=ConstantTimesRule(constant=1/4, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(4*_theta), symbol=_theta), context=cos(4*_theta)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(4*_theta)/2 + 1/2, symbol=_theta), context=cos(2*_theta)**2, symbol=_theta), context=cos(2*_theta)**2/4, symbol=_theta), ConstantTimesRule(constant=1/2, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=cos(2*_theta)/2, symbol=_theta), ConstantRule(constant=1/4, context=1/4, symbol=_theta)], context=cos(2*_theta)**2/4 + cos(2*_theta)/2 + 1/4, symbol=_theta), context=(cos(2*_theta)/2 + 1/2)**2, symbol=_theta), RewriteRule(rewritten=cos(2*_theta)**2/4 + cos(2*_theta)/2 + 1/4, substep=AddRule(substeps=[ConstantTimesRule(constant=1/4, other=cos(2*_theta)**2, substep=RewriteRule(rewritten=cos(4*_theta)/2 + 1/2, substep=AddRule(substeps=[ConstantTimesRule(constant=1/2, other=cos(4*_theta), substep=URule(u_var=_u, u_func=4*_theta, constant=1/4, substep=ConstantTimesRule(constant=1/4, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(4*_theta), symbol=_theta), context=cos(4*_theta)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(4*_theta)/2 + 1/2, symbol=_theta), context=cos(2*_theta)**2, symbol=_theta), context=cos(2*_theta)**2/4, symbol=_theta), ConstantTimesRule(constant=1/2, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=cos(2*_theta)/2, symbol=_theta), ConstantRule(constant=1/4, context=1/4, symbol=_theta)], context=cos(2*_theta)**2/4 + cos(2*_theta)/2 + 1/4, symbol=_theta), context=(cos(2*_theta)/2 + 1/2)**2, symbol=_theta)], context=(cos(2*_theta)/2 + 1/2)**2, symbol=_theta), context=cos(_theta)**4, symbol=_theta), context=sqrt(5)*cos(_theta)**4/125, symbol=_theta), restriction=True, context=(_u**2 + 5)**(-3), symbol=_u)
      
      Por lo tanto, el resultado es: $- \frac{\sqrt{5} \left(\frac{\sqrt{5} u \left(5 - u^{2}\right)}{8 \left(u^{2} + 5\right)^{2}} + \frac{\sqrt{5} u}{2 \left(u^{2} + 5\right)} + \frac{3 \operatorname{atan}{\left(\frac{\sqrt{5} u}{5} \right)}}{8}\right)}{25}$
      
      El resultado es: $\frac{\sqrt{5} \left(\frac{\sqrt{5} u}{2 \left(u^{2} + 5\right)} + \frac{\operatorname{atan}{\left(\frac{\sqrt{5} u}{5} \right)}}{2}\right)}{25} - \frac{\sqrt{5} \left(\frac{\sqrt{5} u \left(5 - u^{2}\right)}{8 \left(u^{2} + 5\right)^{2}} + \frac{\sqrt{5} u}{2 \left(u^{2} + 5\right)} + \frac{3 \operatorname{atan}{\left(\frac{\sqrt{5} u}{5} \right)}}{8}\right)}{25}$
  Por lo tanto, el resultado es: $\frac{2 \sqrt{5} \left(\frac{\sqrt{5} u}{2 \left(u^{2} + 5\right)} + \frac{\operatorname{atan}{\left(\frac{\sqrt{5} u}{5} \right)}}{2}\right)}{25} - \frac{2 \sqrt{5} \left(\frac{\sqrt{5} u \left(5 - u^{2}\right)}{8 \left(u^{2} + 5\right)^{2}} + \frac{\sqrt{5} u}{2 \left(u^{2} + 5\right)} + \frac{3 \operatorname{atan}{\left(\frac{\sqrt{5} u}{5} \right)}}{8}\right)}{25}$
Si ahora sustituir $u$ más en:

$\frac{2 \sqrt{5} \left(\frac{\operatorname{atan}{\left(\frac{\sqrt{5} \sqrt{x - 5}}{5} \right)}}{2} + \frac{\sqrt{5} \sqrt{x - 5}}{2 x}\right)}{25} - \frac{2 \sqrt{5} \left(\frac{3 \operatorname{atan}{\left(\frac{\sqrt{5} \sqrt{x - 5}}{5} \right)}}{8} + \frac{\sqrt{5} \sqrt{x - 5}}{2 x} + \frac{\sqrt{5} \left(10 - x\right) \sqrt{x - 5}}{8 x^{2}}\right)}{25}$
Ahora simplificar:

$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \sqrt{x - 5}}{5} \right)}}{100} + \frac{\sqrt{x - 5}}{20 x} - \frac{\sqrt{x - 5}}{2 x^{2}}$
Añadimos la constante de integración:

$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \sqrt{x - 5}}{5} \right)}}{100} + \frac{\sqrt{x - 5}}{20 x} - \frac{\sqrt{x - 5}}{2 x^{2}}+ \mathrm{constant}$

Respuesta:

$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \sqrt{x - 5}}{5} \right)}}{100} + \frac{\sqrt{x - 5}}{20 x} - \frac{\sqrt{x - 5}}{2 x^{2}}+ \mathrm{constant}$

Respuesta (Indefinida) [src]

                              /      /  ___   _______\                                             \                                                    
                              |      |\/ 5 *\/ x - 5 |                                             |           /    /  ___   _______\                  \
                              |3*atan|---------------|     ___   _______     ___   _______         |           |    |\/ 5 *\/ x - 5 |                  |
  /                       ___ |      \       5       /   \/ 5 *\/ x - 5    \/ 5 *\/ x - 5 *(10 - x)|           |atan|---------------|     ___   _______|
 |                    2*\/ 5 *|----------------------- + --------------- + ------------------------|       ___ |    \       5       /   \/ 5 *\/ x - 5 |
 |   _______                  |           8                    2*x                      2          |   2*\/ 5 *|--------------------- + ---------------|
 | \/ x - 5                   \                                                      8*x           /           \          2                   2*x      /
 | --------- dx = C - ------------------------------------------------------------------------------ + -------------------------------------------------
 |      3                                                   25                                                                 25                       
 |     x                                                                                                                                                
 |                                                                                                                                                      
/

$$\int \frac{\sqrt{x - 5}}{x^{3}}\, dx = C + \frac{2 \sqrt{5} \left(\frac{\operatorname{atan}{\left(\frac{\sqrt{5} \sqrt{x - 5}}{5} \right)}}{2} + \frac{\sqrt{5} \sqrt{x - 5}}{2 x}\right)}{25} - \frac{2 \sqrt{5} \left(\frac{3 \operatorname{atan}{\left(\frac{\sqrt{5} \sqrt{x - 5}}{5} \right)}}{8} + \frac{\sqrt{5} \sqrt{x - 5}}{2 x} + \frac{\sqrt{5} \left(10 - x\right) \sqrt{x - 5}}{8 x^{2}}\right)}{25}$$

Simplificar

Gráfica

Trazar el gráfico f(x)

Respuesta [src]

  1              
  /              
 |               
 |    ________   
 |  \/ -5 + x    
 |  ---------- dx
 |       3       
 |      x        
 |               
/                
0

$$\int\limits_{0}^{1} \frac{\sqrt{x - 5}}{x^{3}}\, dx$$

  1              
  /              
 |               
 |    ________   
 |  \/ -5 + x    
 |  ---------- dx
 |       3       
 |      x        
 |               
/                
0

$$\int\limits_{0}^{1} \frac{\sqrt{x - 5}}{x^{3}}\, dx$$

Integral(sqrt(-5 + x)/x^3, (x, 0, 1))

Respuesta numérica [src]

(0.0 + 2.04681844897888e+38j)

(0.0 + 2.04681844897888e+38j)

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

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Expresiones con funciones

Integral de sqrt(x-5)/x^3 dx

Límites de integración:

Gráfico:

Definida a trozos:

Solución

Método #1

Método #2

Método #3