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Resolución de integrales/ cos3x/ arccos/ x^3*arccos3x

Integral de x^3*arccos3x dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                
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 |   3             
 |  x *acos(3*x) dx
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0
∫01x3acos⁡(3x) dx\int\limits_{0}^{1} x^{3} \operatorname{acos}{\left(3 x \right)}\, dx 
Integral(x^3*acos(3*x), (x, 0, 1))
Solución detallada

  1. Usamos la integración por partes:

    ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

    que u(x)=acos⁡(3x)u{\left(x \right)} = \operatorname{acos}{\left(3 x \right)} y que dv⁡(x)=x3\operatorname{dv}{\left(x \right)} = x^{3}.

    Entonces du⁡(x)=−31−9x2\operatorname{du}{\left(x \right)} = - \frac{3}{\sqrt{1 - 9 x^{2}}}.

    Para buscar v(x)v{\left(x \right)}:

    1. Integral xnx^{n} es xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

      ∫x3 dx=x44\int x^{3}\, dx = \frac{x^{4}}{4}

    Ahora resolvemos podintegral.

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

    ∫(−3x441−9x2) dx=−3∫x41−9x2 dx4\int \left(- \frac{3 x^{4}}{4 \sqrt{1 - 9 x^{2}}}\right)\, dx = - \frac{3 \int \frac{x^{4}}{\sqrt{1 - 9 x^{2}}}\, dx}{4}

      TrigSubstitutionRule(theta=_theta, func=sin(_theta)/3, rewritten=sin(_theta)**4/243, substep=ConstantTimesRule(constant=1/243, other=sin(_theta)**4, substep=RewriteRule(rewritten=(1/2 - cos(2*_theta)/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=(1/2 - cos(2*_theta)/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=(1/2 - cos(2*_theta)/2)**2, symbol=_theta)], context=(1/2 - cos(2*_theta)/2)**2, symbol=_theta), context=sin(_theta)**4, symbol=_theta), context=sin(_theta)**4/243, symbol=_theta), restriction=(x > -1/3) & (x < 1/3), context=x**4/sqrt(1 - 9*x**2), symbol=x)

    Por lo tanto, el resultado es: −3({x(1−18x2)1−9x2648−x1−9x2162+asin⁡(3x)648for x>−13∧x<13)4- \frac{3 \left(\begin{cases} \frac{x \left(1 - 18 x^{2}\right) \sqrt{1 - 9 x^{2}}}{648} - \frac{x \sqrt{1 - 9 x^{2}}}{162} + \frac{\operatorname{asin}{\left(3 x \right)}}{648} & \text{for}\: x > - \frac{1}{3} \wedge x < \frac{1}{3} \end{cases}\right)}{4}

  3. Ahora simplificar:

    {x4acos⁡(3x)4−x31−9x248−x1−9x2288+asin⁡(3x)864for x>−13∧x<13\begin{cases} \frac{x^{4} \operatorname{acos}{\left(3 x \right)}}{4} - \frac{x^{3} \sqrt{1 - 9 x^{2}}}{48} - \frac{x \sqrt{1 - 9 x^{2}}}{288} + \frac{\operatorname{asin}{\left(3 x \right)}}{864} & \text{for}\: x > - \frac{1}{3} \wedge x < \frac{1}{3} \end{cases}

  4. Añadimos la constante de integración:

    {x4acos⁡(3x)4−x31−9x248−x1−9x2288+asin⁡(3x)864for x>−13∧x<13+constant\begin{cases} \frac{x^{4} \operatorname{acos}{\left(3 x \right)}}{4} - \frac{x^{3} \sqrt{1 - 9 x^{2}}}{48} - \frac{x \sqrt{1 - 9 x^{2}}}{288} + \frac{\operatorname{asin}{\left(3 x \right)}}{864} & \text{for}\: x > - \frac{1}{3} \wedge x < \frac{1}{3} \end{cases}+ \mathrm{constant}

Respuesta:

{x4acos⁡(3x)4−x31−9x248−x1−9x2288+asin⁡(3x)864for x>−13∧x<13+constant\begin{cases} \frac{x^{4} \operatorname{acos}{\left(3 x \right)}}{4} - \frac{x^{3} \sqrt{1 - 9 x^{2}}}{48} - \frac{x \sqrt{1 - 9 x^{2}}}{288} + \frac{\operatorname{asin}{\left(3 x \right)}}{864} & \text{for}\: x > - \frac{1}{3} \wedge x < \frac{1}{3} \end{cases}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                           //                 __________        __________                                        \               
                           ||                /        2        /        2  /        2\                            |               
  /                      3*| -1/3, x < 1/3)|    4          
 |  3                      \\   648            162                     648                                        /   x *acos(3*x)
 | x *acos(3*x) dx = C + ------------------------------------------------------------------------------------------ + ------------
 |                                                                   4                                                     4      
/
∫x3acos⁡(3x) dx=C+x4acos⁡(3x)4+3({x(1−18x2)1−9x2648−x1−9x2162+asin⁡(3x)648for x>−13∧x<13)4\int x^{3} \operatorname{acos}{\left(3 x \right)}\, dx = C + \frac{x^{4} \operatorname{acos}{\left(3 x \right)}}{4} + \frac{3 \left(\begin{cases} \frac{x \left(1 - 18 x^{2}\right) \sqrt{1 - 9 x^{2}}}{648} - \frac{x \sqrt{1 - 9 x^{2}}}{162} + \frac{\operatorname{asin}{\left(3 x \right)}}{648} & \text{for}\: x > - \frac{1}{3} \wedge x < \frac{1}{3} \end{cases}\right)}{4}
Simplificar
Gráfica
0.000.050.100.150.200.250.300.02-0.02
Trazar el gráfico f(x)
Respuesta [src] 
                           ___
 pi    215*acos(3)   7*I*\/ 2 
---- + ----------- - ---------
1728       864          144
π1728−72i144+215acos⁡(3)864\frac{\pi}{1728} - \frac{7 \sqrt{2} i}{144} + \frac{215 \operatorname{acos}{\left(3 \right)}}{864} 
=
= 
                           ___
 pi    215*acos(3)   7*I*\/ 2 
---- + ----------- - ---------
1728       864          144
π1728−72i144+215acos⁡(3)864\frac{\pi}{1728} - \frac{7 \sqrt{2} i}{144} + \frac{215 \operatorname{acos}{\left(3 \right)}}{864} 
pi/1728 + 215*acos(3)/864 - 7*i*sqrt(2)/144
Respuesta numérica [src] 
(0.00179673595299029 + 0.369886719553225j)
(0.00179673595299029 + 0.369886719553225j)

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

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