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Resolución de integrales/ sin(xdx/3+4x²)

Integral de sin(xdx/3+4x²) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                 
  /                 
 |                  
 |     /x      2\   
 |  sin|- + 4*x | dx
 |     \3       /   
 |                  
/                   
0
$$\int\limits_{0}^{1} \sin{\left(4 x^{2} + \frac{x}{3} \right)}\, dx$$ 
Integral(sin(x/3 + 4*x^2), (x, 0, 1))
Solución detallada

    FresnelSRule(a=4, b=1/3, c=0, context=sin(4*x**2 + x/3), symbol=x)

  1. Ahora simplificar:

  2. Añadimos la constante de integración:

Respuesta:

Respuesta (Indefinida) [src] 
                                       /            /  ___            \    /  ___            \           \
                            ___   ____ |            |\/ 2 *(1/3 + 8*x)|    |\/ 2 *(1/3 + 8*x)|           |
  /                       \/ 2 *\/ pi *|cos(1/144)*S|-----------------| - C|-----------------|*sin(1/144)|
 |                                     |            |         ____    |    |         ____    |           |
 |    /x      2\                       \            \     4*\/ pi     /    \     4*\/ pi     /           /
 | sin|- + 4*x | dx = C + --------------------------------------------------------------------------------
 |    \3       /                                                 4                                        
 |                                                                                                        
/
$$\int \sin{\left(4 x^{2} + \frac{x}{3} \right)}\, dx = C + \frac{\sqrt{2} \sqrt{\pi} \left(- \sin{\left(\frac{1}{144} \right)} C\left(\frac{\sqrt{2} \left(8 x + \frac{1}{3}\right)}{4 \sqrt{\pi}}\right) + \cos{\left(\frac{1}{144} \right)} S\left(\frac{\sqrt{2} \left(8 x + \frac{1}{3}\right)}{4 \sqrt{\pi}}\right)\right)}{4}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
               /            /    ___  \    /    ___  \           \                /            /      ___\    /      ___\           \
    ___   ____ |            |  \/ 2   |    |  \/ 2   |           |     ___   ____ |            | 25*\/ 2 |    | 25*\/ 2 |           |
  \/ 2 *\/ pi *|cos(1/144)*S|---------| - C|---------|*sin(1/144)|   \/ 2 *\/ pi *|cos(1/144)*S|---------| - C|---------|*sin(1/144)|
               |            |     ____|    |     ____|           |                |            |     ____|    |     ____|           |
               \            \12*\/ pi /    \12*\/ pi /           /                \            \12*\/ pi /    \12*\/ pi /           /
- ---------------------------------------------------------------- + ----------------------------------------------------------------
                                 4                                                                  4
$$- \frac{\sqrt{2} \sqrt{\pi} \left(- \sin{\left(\frac{1}{144} \right)} C\left(\frac{\sqrt{2}}{12 \sqrt{\pi}}\right) + \cos{\left(\frac{1}{144} \right)} S\left(\frac{\sqrt{2}}{12 \sqrt{\pi}}\right)\right)}{4} + \frac{\sqrt{2} \sqrt{\pi} \left(- \sin{\left(\frac{1}{144} \right)} C\left(\frac{25 \sqrt{2}}{12 \sqrt{\pi}}\right) + \cos{\left(\frac{1}{144} \right)} S\left(\frac{25 \sqrt{2}}{12 \sqrt{\pi}}\right)\right)}{4}$$ 
=
= 
               /            /    ___  \    /    ___  \           \                /            /      ___\    /      ___\           \
    ___   ____ |            |  \/ 2   |    |  \/ 2   |           |     ___   ____ |            | 25*\/ 2 |    | 25*\/ 2 |           |
  \/ 2 *\/ pi *|cos(1/144)*S|---------| - C|---------|*sin(1/144)|   \/ 2 *\/ pi *|cos(1/144)*S|---------| - C|---------|*sin(1/144)|
               |            |     ____|    |     ____|           |                |            |     ____|    |     ____|           |
               \            \12*\/ pi /    \12*\/ pi /           /                \            \12*\/ pi /    \12*\/ pi /           /
- ---------------------------------------------------------------- + ----------------------------------------------------------------
                                 4                                                                  4
$$- \frac{\sqrt{2} \sqrt{\pi} \left(- \sin{\left(\frac{1}{144} \right)} C\left(\frac{\sqrt{2}}{12 \sqrt{\pi}}\right) + \cos{\left(\frac{1}{144} \right)} S\left(\frac{\sqrt{2}}{12 \sqrt{\pi}}\right)\right)}{4} + \frac{\sqrt{2} \sqrt{\pi} \left(- \sin{\left(\frac{1}{144} \right)} C\left(\frac{25 \sqrt{2}}{12 \sqrt{\pi}}\right) + \cos{\left(\frac{1}{144} \right)} S\left(\frac{25 \sqrt{2}}{12 \sqrt{\pi}}\right)\right)}{4}$$ 
-sqrt(2)*sqrt(pi)*(cos(1/144)*fresnels(sqrt(2)/(12*sqrt(pi))) - fresnelc(sqrt(2)/(12*sqrt(pi)))*sin(1/144))/4 + sqrt(2)*sqrt(pi)*(cos(1/144)*fresnels(25*sqrt(2)/(12*sqrt(pi))) - fresnelc(25*sqrt(2)/(12*sqrt(pi)))*sin(1/144))/4
Respuesta numérica [src] 
0.365626115430716
0.365626115430716

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

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