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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

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

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

  1. Añadimos la constante de integración:

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

Respuesta:

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

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

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

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