Sr Examen

Otras calculadoras

Resolución de integrales/ 6*x^2/ x^2+1/ x^3/sqrt(16*x^2+1)

Integral de x^3/sqrt(16*x^2+1) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                  
  /                  
 |                   
 |         3         
 |        x          
 |  -------------- dx
 |     ___________   
 |    /     2        
 |  \/  16*x  + 1    
 |                   
/                    
0
$$\int\limits_{0}^{1} \frac{x^{3}}{\sqrt{16 x^{2} + 1}}\, dx$$ 
Integral(x^3/sqrt(16*x^2 + 1), (x, 0, 1))
Solución detallada

    TrigSubstitutionRule(theta=_theta, func=tan(_theta)/4, rewritten=sin(_theta)**3/(256*cos(_theta)**4), substep=ConstantTimesRule(constant=1/256, other=sin(_theta)**3/cos(_theta)**4, substep=RewriteRule(rewritten=(1 - cos(_theta)**2)*sin(_theta)/cos(_theta)**4, substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=cos(_theta), constant=1, substep=RewriteRule(rewritten=_u**(-2) - 1/_u**4, substep=AddRule(substeps=[PowerRule(base=_u, exp=-2, context=_u**(-2), symbol=_u), ConstantTimesRule(constant=-1, other=_u**(-4), substep=PowerRule(base=_u, exp=-4, context=_u**(-4), symbol=_u), context=-1/_u**4, symbol=_u)], context=_u**(-2) - 1/_u**4, symbol=_u), context=(_u**2 - 1)/_u**4, symbol=_u), context=(1 - cos(_theta)**2)*sin(_theta)/cos(_theta)**4, symbol=_theta), RewriteRule(rewritten=-(sin(_theta)*cos(_theta)**2 - sin(_theta))/cos(_theta)**4, substep=ConstantTimesRule(constant=-1, other=(sin(_theta)*cos(_theta)**2 - sin(_theta))/cos(_theta)**4, substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=cos(_theta), constant=-1, substep=ConstantTimesRule(constant=-1, other=(_u**2 - 1)/_u**4, substep=RewriteRule(rewritten=_u**(-2) - 1/_u**4, substep=AddRule(substeps=[PowerRule(base=_u, exp=-2, context=_u**(-2), symbol=_u), ConstantTimesRule(constant=-1, other=_u**(-4), substep=PowerRule(base=_u, exp=-4, context=_u**(-4), symbol=_u), context=-1/_u**4, symbol=_u)], context=_u**(-2) - 1/_u**4, symbol=_u), context=(_u**2 - 1)/_u**4, symbol=_u), context=(_u**2 - 1)/_u**4, symbol=_u), context=(sin(_theta)*cos(_theta)**2 - sin(_theta))/cos(_theta)**4, symbol=_theta), RewriteRule(rewritten=sin(_theta)/cos(_theta)**2 - sin(_theta)/cos(_theta)**4, substep=AddRule(substeps=[URule(u_var=_u, u_func=cos(_theta), constant=-1, substep=ConstantTimesRule(constant=-1, other=_u**(-2), substep=PowerRule(base=_u, exp=-2, context=_u**(-2), symbol=_u), context=_u**(-2), symbol=_u), context=sin(_theta)/cos(_theta)**2, symbol=_theta), ConstantTimesRule(constant=-1, other=sin(_theta)/cos(_theta)**4, substep=URule(u_var=_u, u_func=cos(_theta), constant=-1, substep=ConstantTimesRule(constant=-1, other=_u**(-4), substep=PowerRule(base=_u, exp=-4, context=_u**(-4), symbol=_u), context=_u**(-4), symbol=_u), context=sin(_theta)/cos(_theta)**4, symbol=_theta), context=-sin(_theta)/cos(_theta)**4, symbol=_theta)], context=sin(_theta)/cos(_theta)**2 - sin(_theta)/cos(_theta)**4, symbol=_theta), context=(sin(_theta)*cos(_theta)**2 - sin(_theta))/cos(_theta)**4, symbol=_theta)], context=(sin(_theta)*cos(_theta)**2 - sin(_theta))/cos(_theta)**4, symbol=_theta), context=-(sin(_theta)*cos(_theta)**2 - sin(_theta))/cos(_theta)**4, symbol=_theta), context=(1 - cos(_theta)**2)*sin(_theta)/cos(_theta)**4, symbol=_theta), RewriteRule(rewritten=-sin(_theta)/cos(_theta)**2 + sin(_theta)/cos(_theta)**4, substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=sin(_theta)/cos(_theta)**2, substep=URule(u_var=_u, u_func=cos(_theta), constant=-1, substep=ConstantTimesRule(constant=-1, other=_u**(-2), substep=PowerRule(base=_u, exp=-2, context=_u**(-2), symbol=_u), context=_u**(-2), symbol=_u), context=sin(_theta)/cos(_theta)**2, symbol=_theta), context=-sin(_theta)/cos(_theta)**2, symbol=_theta), URule(u_var=_u, u_func=cos(_theta), constant=-1, substep=ConstantTimesRule(constant=-1, other=_u**(-4), substep=PowerRule(base=_u, exp=-4, context=_u**(-4), symbol=_u), context=_u**(-4), symbol=_u), context=sin(_theta)/cos(_theta)**4, symbol=_theta)], context=-sin(_theta)/cos(_theta)**2 + sin(_theta)/cos(_theta)**4, symbol=_theta), context=(1 - cos(_theta)**2)*sin(_theta)/cos(_theta)**4, symbol=_theta)], context=(1 - cos(_theta)**2)*sin(_theta)/cos(_theta)**4, symbol=_theta), context=sin(_theta)**3/cos(_theta)**4, symbol=_theta), context=sin(_theta)**3/(256*cos(_theta)**4), symbol=_theta), restriction=True, context=x**3/sqrt(16*x**2 + 1), symbol=x)

  1. Ahora simplificar:

  2. Añadimos la constante de integración:

Respuesta:

Respuesta (Indefinida) [src] 
  /                                                       
 |                            ___________              3/2
 |        3                  /         2    /        2\   
 |       x                 \/  1 + 16*x     \1 + 16*x /   
 | -------------- dx = C - -------------- + --------------
 |    ___________               256              768      
 |   /     2                                              
 | \/  16*x  + 1                                          
 |                                                        
/
$$\int \frac{x^{3}}{\sqrt{16 x^{2} + 1}}\, dx = C + \frac{\left(16 x^{2} + 1\right)^{\frac{3}{2}}}{768} - \frac{\sqrt{16 x^{2} + 1}}{256}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
          ____
 1    7*\/ 17 
--- + --------
384     384
$$\frac{1}{384} + \frac{7 \sqrt{17}}{384}$$ 
=
= 
          ____
 1    7*\/ 17 
--- + --------
384     384
$$\frac{1}{384} + \frac{7 \sqrt{17}}{384}$$ 
1/384 + 7*sqrt(17)/384
Respuesta numérica [src] 
0.0777649463003219
0.0777649463003219

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

    © Sr Examen – Калькулятор