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Resolución de integrales/ x^2+9/ 1/(x^2+9)^3/2

Integral de 1/(x^2+9)^3/2 dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1               
  /               
 |                
 |       1        
 |  ----------- dx
 |          3     
 |  / 2    \      
 |  \x  + 9/ *2   
 |                
/                 
0
∫0112(x2+9)3 dx\int\limits_{0}^{1} \frac{1}{2 \left(x^{2} + 9\right)^{3}}\, dx 
Integral(1/((x^2 + 9)^3*2), (x, 0, 1))
Solución detallada

  1. La integral del producto de una función por una constante es la constante por la integral de esta función:

    ∫12(x2+9)3 dx=∫1(x2+9)3 dx2\int \frac{1}{2 \left(x^{2} + 9\right)^{3}}\, dx = \frac{\int \frac{1}{\left(x^{2} + 9\right)^{3}}\, dx}{2}

      TrigSubstitutionRule(theta=_theta, func=3*tan(_theta), rewritten=cos(_theta)**4/243, substep=ConstantTimesRule(constant=1/243, 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=cos(_theta)**4/243, symbol=_theta), restriction=True, context=1/((x**2 + 9)**3), symbol=x)

    Por lo tanto, el resultado es: x(9−x2)1296(x2+9)2+x324(x2+9)+atan⁡(x3)1296\frac{x \left(9 - x^{2}\right)}{1296 \left(x^{2} + 9\right)^{2}} + \frac{x}{324 \left(x^{2} + 9\right)} + \frac{\operatorname{atan}{\left(\frac{x}{3} \right)}}{1296}

  2. Ahora simplificar:

    x3432(x2+9)2+5x144(x2+9)2+atan⁡(x3)1296\frac{x^{3}}{432 \left(x^{2} + 9\right)^{2}} + \frac{5 x}{144 \left(x^{2} + 9\right)^{2}} + \frac{\operatorname{atan}{\left(\frac{x}{3} \right)}}{1296}

  3. Añadimos la constante de integración:

    x3432(x2+9)2+5x144(x2+9)2+atan⁡(x3)1296+constant\frac{x^{3}}{432 \left(x^{2} + 9\right)^{2}} + \frac{5 x}{144 \left(x^{2} + 9\right)^{2}} + \frac{\operatorname{atan}{\left(\frac{x}{3} \right)}}{1296}+ \mathrm{constant}

Respuesta:

x3432(x2+9)2+5x144(x2+9)2+atan⁡(x3)1296+constant\frac{x^{3}}{432 \left(x^{2} + 9\right)^{2}} + \frac{5 x}{144 \left(x^{2} + 9\right)^{2}} + \frac{\operatorname{atan}{\left(\frac{x}{3} \right)}}{1296}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
  /                         /x\                                
 |                      atan|-|                      /     2\  
 |      1                   \3/        x           x*\9 - x /  
 | ----------- dx = C + ------- + ------------ + --------------
 |         3              1296        /     2\                2
 | / 2    \                       324*\9 + x /        /     2\ 
 | \x  + 9/ *2                                   1296*\9 + x / 
 |                                                             
/
∫12(x2+9)3 dx=C+x(9−x2)1296(x2+9)2+x324(x2+9)+atan⁡(x3)1296\int \frac{1}{2 \left(x^{2} + 9\right)^{3}}\, dx = C + \frac{x \left(9 - x^{2}\right)}{1296 \left(x^{2} + 9\right)^{2}} + \frac{x}{324 \left(x^{2} + 9\right)} + \frac{\operatorname{atan}{\left(\frac{x}{3} \right)}}{1296}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.900.00000.0010
Trazar el gráfico f(x)
Respuesta [src] 
 1     atan(1/3)
---- + ---------
2700      1296
atan⁡(13)1296+12700\frac{\operatorname{atan}{\left(\frac{1}{3} \right)}}{1296} + \frac{1}{2700} 
=
= 
 1     atan(1/3)
---- + ---------
2700      1296
atan⁡(13)1296+12700\frac{\operatorname{atan}{\left(\frac{1}{3} \right)}}{1296} + \frac{1}{2700} 
1/2700 + atan(1/3)/1296
Respuesta numérica [src] 
0.000618634687034446
0.000618634687034446

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

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