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Resolución de integrales/ x^2+2/ y^2+1/ sin(2x^2+2y^2+1)

Integral de sin(2x^2+2y^2+1) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                        
  /                        
 |                         
 |     /   2      2    \   
 |  sin\2*x  + 2*y  + 1/ dx
 |                         
/                          
0
∫01sin⁡((2x2+2y2)+1) dx\int\limits_{0}^{1} \sin{\left(\left(2 x^{2} + 2 y^{2}\right) + 1 \right)}\, dx 
Integral(sin(2*x^2 + 2*y^2 + 1), (x, 0, 1))
Solución detallada

    FresnelSRule(a=2, b=0, c=2*y**2 + 1, context=sin(2*x**2 + 2*y**2 + 1), symbol=x)

  1. Añadimos la constante de integración:

    π(sin⁡(2y2+1)C(2xπ)+cos⁡(2y2+1)S(2xπ))2+constant\frac{\sqrt{\pi} \left(\sin{\left(2 y^{2} + 1 \right)} C\left(\frac{2 x}{\sqrt{\pi}}\right) + \cos{\left(2 y^{2} + 1 \right)} S\left(\frac{2 x}{\sqrt{\pi}}\right)\right)}{2}+ \mathrm{constant}

Respuesta:

π(sin⁡(2y2+1)C(2xπ)+cos⁡(2y2+1)S(2xπ))2+constant\frac{\sqrt{\pi} \left(\sin{\left(2 y^{2} + 1 \right)} C\left(\frac{2 x}{\sqrt{\pi}}\right) + \cos{\left(2 y^{2} + 1 \right)} S\left(\frac{2 x}{\sqrt{\pi}}\right)\right)}{2}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                                   ____ /   /       2\  / 2*x  \    / 2*x  \    /       2\\
  /                              \/ pi *|cos\1 + 2*y /*S|------| + C|------|*sin\1 + 2*y /|
 |                                      |               |  ____|    |  ____|              |
 |    /   2      2    \                 \               \\/ pi /    \\/ pi /              /
 | sin\2*x  + 2*y  + 1/ dx = C + ----------------------------------------------------------
 |                                                           2                             
/
∫sin⁡((2x2+2y2)+1) dx=C+π(sin⁡(2y2+1)C(2xπ)+cos⁡(2y2+1)S(2xπ))2\int \sin{\left(\left(2 x^{2} + 2 y^{2}\right) + 1 \right)}\, dx = C + \frac{\sqrt{\pi} \left(\sin{\left(2 y^{2} + 1 \right)} C\left(\frac{2 x}{\sqrt{\pi}}\right) + \cos{\left(2 y^{2} + 1 \right)} S\left(\frac{2 x}{\sqrt{\pi}}\right)\right)}{2}
Simplificar
Respuesta [src] 
  ____ /   /       2\  /  2   \    /  2   \    /       2\\
\/ pi *|cos\1 + 2*y /*S|------| + C|------|*sin\1 + 2*y /|
       |               |  ____|    |  ____|              |
       \               \\/ pi /    \\/ pi /              /
----------------------------------------------------------
                            2
π(sin⁡(2y2+1)C(2π)+cos⁡(2y2+1)S(2π))2\frac{\sqrt{\pi} \left(\sin{\left(2 y^{2} + 1 \right)} C\left(\frac{2}{\sqrt{\pi}}\right) + \cos{\left(2 y^{2} + 1 \right)} S\left(\frac{2}{\sqrt{\pi}}\right)\right)}{2} 
=
= 
  ____ /   /       2\  /  2   \    /  2   \    /       2\\
\/ pi *|cos\1 + 2*y /*S|------| + C|------|*sin\1 + 2*y /|
       |               |  ____|    |  ____|              |
       \               \\/ pi /    \\/ pi /              /
----------------------------------------------------------
                            2
π(sin⁡(2y2+1)C(2π)+cos⁡(2y2+1)S(2π))2\frac{\sqrt{\pi} \left(\sin{\left(2 y^{2} + 1 \right)} C\left(\frac{2}{\sqrt{\pi}}\right) + \cos{\left(2 y^{2} + 1 \right)} S\left(\frac{2}{\sqrt{\pi}}\right)\right)}{2} 
sqrt(pi)*(cos(1 + 2*y^2)*fresnels(2/sqrt(pi)) + fresnelc(2/sqrt(pi))*sin(1 + 2*y^2))/2

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

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