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Resolución de integrales/ sqrt((2sin2x-2sinx)^2+(2cosx-2cos2x)^2)

Integral de sqrt((2sin2x-2sinx)^2+(2cosx-2cos2x)^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 pi                                                            
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 |  \/  (2*sin(2*x) - 2*sin(x))  + (2*cos(x) - 2*cos(2*x))   dx
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0
∫0π(−2sin⁡(x)+2sin⁡(2x))2+(2cos⁡(x)−2cos⁡(2x))2 dx\int\limits_{0}^{\pi} \sqrt{\left(- 2 \sin{\left(x \right)} + 2 \sin{\left(2 x \right)}\right)^{2} + \left(2 \cos{\left(x \right)} - 2 \cos{\left(2 x \right)}\right)^{2}}\, dx 
Integral(sqrt((2*sin(2*x) - 2*sin(x))^2 + (2*cos(x) - 2*cos(2*x))^2), (x, 0, pi))
Respuesta (Indefinida) [src] 
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 |   /                        2                          2              |   /    2         2           2         2                                                 
 | \/  (2*sin(2*x) - 2*sin(x))  + (2*cos(x) - 2*cos(2*x))   dx = C + 2* | \/  cos (x) + cos (2*x) + sin (x) + sin (2*x) - 2*cos(x)*cos(2*x) - 2*sin(x)*sin(2*x)  dx
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∫(−2sin⁡(x)+2sin⁡(2x))2+(2cos⁡(x)−2cos⁡(2x))2 dx=C+2∫sin⁡2(x)−2sin⁡(x)sin⁡(2x)+sin⁡2(2x)+cos⁡2(x)−2cos⁡(x)cos⁡(2x)+cos⁡2(2x) dx\int \sqrt{\left(- 2 \sin{\left(x \right)} + 2 \sin{\left(2 x \right)}\right)^{2} + \left(2 \cos{\left(x \right)} - 2 \cos{\left(2 x \right)}\right)^{2}}\, dx = C + 2 \int \sqrt{\sin^{2}{\left(x \right)} - 2 \sin{\left(x \right)} \sin{\left(2 x \right)} + \sin^{2}{\left(2 x \right)} + \cos^{2}{\left(x \right)} - 2 \cos{\left(x \right)} \cos{\left(2 x \right)} + \cos^{2}{\left(2 x \right)}}\, dx
Simplificar
Respuesta [src] 
   pi                                                                                          
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   |    /    2         2           2         2                                                 
2* |  \/  cos (x) + cos (2*x) + sin (x) + sin (2*x) - 2*cos(x)*cos(2*x) - 2*sin(x)*sin(2*x)  dx
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  0
2∫0πsin⁡2(x)−2sin⁡(x)sin⁡(2x)+sin⁡2(2x)+cos⁡2(x)−2cos⁡(x)cos⁡(2x)+cos⁡2(2x) dx2 \int\limits_{0}^{\pi} \sqrt{\sin^{2}{\left(x \right)} - 2 \sin{\left(x \right)} \sin{\left(2 x \right)} + \sin^{2}{\left(2 x \right)} + \cos^{2}{\left(x \right)} - 2 \cos{\left(x \right)} \cos{\left(2 x \right)} + \cos^{2}{\left(2 x \right)}}\, dx 
=
= 
   pi                                                                                          
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   |    /    2         2           2         2                                                 
2* |  \/  cos (x) + cos (2*x) + sin (x) + sin (2*x) - 2*cos(x)*cos(2*x) - 2*sin(x)*sin(2*x)  dx
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  0
2∫0πsin⁡2(x)−2sin⁡(x)sin⁡(2x)+sin⁡2(2x)+cos⁡2(x)−2cos⁡(x)cos⁡(2x)+cos⁡2(2x) dx2 \int\limits_{0}^{\pi} \sqrt{\sin^{2}{\left(x \right)} - 2 \sin{\left(x \right)} \sin{\left(2 x \right)} + \sin^{2}{\left(2 x \right)} + \cos^{2}{\left(x \right)} - 2 \cos{\left(x \right)} \cos{\left(2 x \right)} + \cos^{2}{\left(2 x \right)}}\, dx 
2*Integral(sqrt(cos(x)^2 + cos(2*x)^2 + sin(x)^2 + sin(2*x)^2 - 2*cos(x)*cos(2*x) - 2*sin(x)*sin(2*x)), (x, 0, pi))
Respuesta numérica [src] 
8.0
8.0

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

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