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Resolución de integrales/ x^2+5/ x^5sqrt(x^2+5)dx

Integral de x^5sqrt(x^2+5)dx dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                  
  /                  
 |                   
 |        ________   
 |   5   /  2        
 |  x *\/  x  + 5  dx
 |                   
/                    
0
∫01x5x2+5 dx\int\limits_{0}^{1} x^{5} \sqrt{x^{2} + 5}\, dx 
Integral(x^5*sqrt(x^2 + 5), (x, 0, 1))
Solución detallada

    TrigSubstitutionRule(theta=_theta, func=sqrt(5)*tan(_theta), rewritten=125*sqrt(5)*tan(_theta)**5/cos(_theta)**3, substep=ConstantTimesRule(constant=125*sqrt(5), other=tan(_theta)**5/cos(_theta)**3, substep=RewriteRule(rewritten=(sec(_theta)**2 - 1)**2*tan(_theta)*sec(_theta)**3, substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=sec(_theta), constant=1, substep=AddRule(substeps=[PowerRule(base=_u, exp=6, context=_u**6, symbol=_u), ConstantTimesRule(constant=-2, other=_u**4, substep=PowerRule(base=_u, exp=4, context=_u**4, symbol=_u), context=-2*_u**4, symbol=_u), PowerRule(base=_u, exp=2, context=_u**2, symbol=_u)], context=_u**6 - 2*_u**4 + _u**2, symbol=_u), context=(sec(_theta)**2 - 1)**2*tan(_theta)*sec(_theta)**3, symbol=_theta), RewriteRule(rewritten=tan(_theta)*sec(_theta)**7 - 2*tan(_theta)*sec(_theta)**5 + tan(_theta)*sec(_theta)**3, substep=AddRule(substeps=[URule(u_var=_u, u_func=sec(_theta), constant=1, substep=PowerRule(base=_u, exp=6, context=_u**6, symbol=_u), context=tan(_theta)*sec(_theta)**7, symbol=_theta), ConstantTimesRule(constant=-2, other=tan(_theta)*sec(_theta)**5, substep=URule(u_var=_u, u_func=sec(_theta), constant=1, substep=PowerRule(base=_u, exp=4, context=_u**4, symbol=_u), context=tan(_theta)*sec(_theta)**5, symbol=_theta), context=-2*tan(_theta)*sec(_theta)**5, symbol=_theta), URule(u_var=_u, u_func=sec(_theta), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=tan(_theta)*sec(_theta)**3, symbol=_theta)], context=tan(_theta)*sec(_theta)**7 - 2*tan(_theta)*sec(_theta)**5 + tan(_theta)*sec(_theta)**3, symbol=_theta), context=(sec(_theta)**2 - 1)**2*tan(_theta)*sec(_theta)**3, symbol=_theta), RewriteRule(rewritten=tan(_theta)*sec(_theta)**7 - 2*tan(_theta)*sec(_theta)**5 + tan(_theta)*sec(_theta)**3, substep=AddRule(substeps=[URule(u_var=_u, u_func=sec(_theta), constant=1, substep=PowerRule(base=_u, exp=6, context=_u**6, symbol=_u), context=tan(_theta)*sec(_theta)**7, symbol=_theta), ConstantTimesRule(constant=-2, other=tan(_theta)*sec(_theta)**5, substep=URule(u_var=_u, u_func=sec(_theta), constant=1, substep=PowerRule(base=_u, exp=4, context=_u**4, symbol=_u), context=tan(_theta)*sec(_theta)**5, symbol=_theta), context=-2*tan(_theta)*sec(_theta)**5, symbol=_theta), URule(u_var=_u, u_func=sec(_theta), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=tan(_theta)*sec(_theta)**3, symbol=_theta)], context=tan(_theta)*sec(_theta)**7 - 2*tan(_theta)*sec(_theta)**5 + tan(_theta)*sec(_theta)**3, symbol=_theta), context=(sec(_theta)**2 - 1)**2*tan(_theta)*sec(_theta)**3, symbol=_theta)], context=(sec(_theta)**2 - 1)**2*tan(_theta)*sec(_theta)**3, symbol=_theta), context=tan(_theta)**5*sec(_theta)**3, symbol=_theta), context=125*sqrt(5)*tan(_theta)**5/cos(_theta)**3, symbol=_theta), restriction=True, context=x**5*sqrt(x**2 + 5), symbol=x)

  1. Ahora simplificar:

    (x2+5)32(−42x2+3(x2+5)2−35)21\frac{\left(x^{2} + 5\right)^{\frac{3}{2}} \left(- 42 x^{2} + 3 \left(x^{2} + 5\right)^{2} - 35\right)}{21}

  2. Añadimos la constante de integración:

    (x2+5)32(−42x2+3(x2+5)2−35)21+constant\frac{\left(x^{2} + 5\right)^{\frac{3}{2}} \left(- 42 x^{2} + 3 \left(x^{2} + 5\right)^{2} - 35\right)}{21}+ \mathrm{constant}

Respuesta:

(x2+5)32(−42x2+3(x2+5)2−35)21+constant\frac{\left(x^{2} + 5\right)^{\frac{3}{2}} \left(- 42 x^{2} + 3 \left(x^{2} + 5\right)^{2} - 35\right)}{21}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                                     /            5/2           3/2           7/2\
  /                                  |    /     2\      /     2\      /     2\   |
 |                                   |    |    x |      |    x |      |    x |   |
 |       ________                    |  2*|1 + --|      |1 + --|      |1 + --|   |
 |  5   /  2                     ___ |    \    5 /      \    5 /      \    5 /   |
 | x *\/  x  + 5  dx = C + 125*\/ 5 *|- ------------- + ----------- + -----------|
 |                                   \        5              3             7     /
/
∫x5x2+5 dx=C+1255((x25+1)727−2(x25+1)525+(x25+1)323)\int x^{5} \sqrt{x^{2} + 5}\, dx = C + 125 \sqrt{5} \left(\frac{\left(\frac{x^{2}}{5} + 1\right)^{\frac{7}{2}}}{7} - \frac{2 \left(\frac{x^{2}}{5} + 1\right)^{\frac{5}{2}}}{5} + \frac{\left(\frac{x^{2}}{5} + 1\right)^{\frac{3}{2}}}{3}\right)
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.90025
Trazar el gráfico f(x)
Respuesta [src] 
        ___        ___
  200*\/ 5    62*\/ 6 
- --------- + --------
      21         7
−200521+6267- \frac{200 \sqrt{5}}{21} + \frac{62 \sqrt{6}}{7} 
=
= 
        ___        ___
  200*\/ 5    62*\/ 6 
- --------- + --------
      21         7
−200521+6267- \frac{200 \sqrt{5}}{21} + \frac{62 \sqrt{6}}{7} 
-200*sqrt(5)/21 + 62*sqrt(6)/7
Respuesta numérica [src] 
0.399595078938723
0.399595078938723

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

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