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Resolución de integrales/ 5-x^2/ sqrt(25-x^2)+sqrt(25-x^2)

Integral de sqrt(25-x^2)+sqrt(25-x^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 10                                 
  /                                 
 |                                  
 |  /   _________      _________\   
 |  |  /       2      /       2 |   
 |  \\/  25 - x   + \/  25 - x  / dx
 |                                  
/                                   
-10
∫−1010(25−x2+25−x2) dx\int\limits_{-10}^{10} \left(\sqrt{25 - x^{2}} + \sqrt{25 - x^{2}}\right)\, dx 
Integral(sqrt(25 - x^2) + sqrt(25 - x^2), (x, -10, 10))
Solución detallada

  1. Integramos término a término:

      TrigSubstitutionRule(theta=_theta, func=5*sin(_theta), rewritten=25*cos(_theta)**2, substep=ConstantTimesRule(constant=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=25*cos(_theta)**2, symbol=_theta), restriction=(x > -5) & (x < 5), context=sqrt(25 - x**2), symbol=x)

      TrigSubstitutionRule(theta=_theta, func=5*sin(_theta), rewritten=25*cos(_theta)**2, substep=ConstantTimesRule(constant=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=25*cos(_theta)**2, symbol=_theta), restriction=(x > -5) & (x < 5), context=sqrt(25 - x**2), symbol=x)

    El resultado es: 2({x25−x22+25asin⁡(x5)2for x>−5∧x<5)2 \left(\begin{cases} \frac{x \sqrt{25 - x^{2}}}{2} + \frac{25 \operatorname{asin}{\left(\frac{x}{5} \right)}}{2} & \text{for}\: x > -5 \wedge x < 5 \end{cases}\right)

  2. Ahora simplificar:

    {x25−x2+25asin⁡(x5)for x>−5∧x<5\begin{cases} x \sqrt{25 - x^{2}} + 25 \operatorname{asin}{\left(\frac{x}{5} \right)} & \text{for}\: x > -5 \wedge x < 5 \end{cases}

  3. Añadimos la constante de integración:

    {x25−x2+25asin⁡(x5)for x>−5∧x<5+constant\begin{cases} x \sqrt{25 - x^{2}} + 25 \operatorname{asin}{\left(\frac{x}{5} \right)} & \text{for}\: x > -5 \wedge x < 5 \end{cases}+ \mathrm{constant}

Respuesta:

{x25−x2+25asin⁡(x5)for x>−5∧x<5+constant\begin{cases} x \sqrt{25 - x^{2}} + 25 \operatorname{asin}{\left(\frac{x}{5} \right)} & \text{for}\: x > -5 \wedge x < 5 \end{cases}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
  /                                                                                               
 |                                                                                                
 | /   _________      _________\            //       /x\        _________                        \
 | |  /       2      /       2 |            ||25*asin|-|       /       2                         |
 | \\/  25 - x   + \/  25 - x  / dx = C + 2*|<       \5/   x*\/  25 - x                          |
 |                                          ||---------- + --------------  for And(x > -5, x < 5)|
/                                           \\    2              2                               /
∫(25−x2+25−x2) dx=C+2({x25−x22+25asin⁡(x5)2for x>−5∧x<5)\int \left(\sqrt{25 - x^{2}} + \sqrt{25 - x^{2}}\right)\, dx = C + 2 \left(\begin{cases} \frac{x \sqrt{25 - x^{2}}}{2} + \frac{25 \operatorname{asin}{\left(\frac{x}{5} \right)}}{2} & \text{for}\: x > -5 \wedge x < 5 \end{cases}\right)
Simplificar
Gráfica
-5.0-4.0-3.0-2.0-1.00.01.02.03.04.05.0-100100
Trazar el gráfico f(x)
Respuesta [src] 
                     ___
50*asin(2) + 100*I*\/ 3
50asin⁡(2)+1003i50 \operatorname{asin}{\left(2 \right)} + 100 \sqrt{3} i 
=
= 
                     ___
50*asin(2) + 100*I*\/ 3
50asin⁡(2)+1003i50 \operatorname{asin}{\left(2 \right)} + 100 \sqrt{3} i 
50*asin(2) + 100*i*sqrt(3)
Respuesta numérica [src] 
(78.3535520096473 + 107.337370499741j)
(78.3535520096473 + 107.337370499741j)

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

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