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Resolución de integrales/ sin(x)/ (x^2)/ sin(x)*sin(x^2)

Integral de sin(x)*sin(x^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 oo                  
  /                  
 |                   
 |            / 2\   
 |  sin(x)*sin\x / dx
 |                   
/                    
1
∫1∞sin⁡(x)sin⁡(x2) dx\int\limits_{1}^{\infty} \sin{\left(x \right)} \sin{\left(x^{2} \right)}\, dx 
Integral(sin(x)*sin(x^2), (x, 1, oo))
Respuesta (Indefinida) [src] 
                                        /          /  ___          \    /  ___          \         \                /          /  ___           \    /  ___           \         \
                             ___   ____ |          |\/ 2 *(1 + 2*x)|    |\/ 2 *(1 + 2*x)|         |     ___   ____ |          |\/ 2 *(-1 + 2*x)|    |\/ 2 *(-1 + 2*x)|         |
  /                        \/ 2 *\/ pi *|cos(1/4)*C|---------------| + S|---------------|*sin(1/4)|   \/ 2 *\/ pi *|cos(1/4)*C|----------------| + S|----------------|*sin(1/4)|
 |                                      |          |        ____   |    |        ____   |         |                |          |        ____    |    |        ____    |         |
 |           / 2\                       \          \    2*\/ pi    /    \    2*\/ pi    /         /                \          \    2*\/ pi     /    \    2*\/ pi     /         /
 | sin(x)*sin\x / dx = C - ------------------------------------------------------------------------ + --------------------------------------------------------------------------
 |                                                            4                                                                           4                                     
/
∫sin⁡(x)sin⁡(x2) dx=C+2π(cos⁡(14)C(2(2x−1)2π)+sin⁡(14)S(2(2x−1)2π))4−2π(cos⁡(14)C(2(2x+1)2π)+sin⁡(14)S(2(2x+1)2π))4\int \sin{\left(x \right)} \sin{\left(x^{2} \right)}\, dx = C + \frac{\sqrt{2} \sqrt{\pi} \left(\cos{\left(\frac{1}{4} \right)} C\left(\frac{\sqrt{2} \left(2 x - 1\right)}{2 \sqrt{\pi}}\right) + \sin{\left(\frac{1}{4} \right)} S\left(\frac{\sqrt{2} \left(2 x - 1\right)}{2 \sqrt{\pi}}\right)\right)}{4} - \frac{\sqrt{2} \sqrt{\pi} \left(\cos{\left(\frac{1}{4} \right)} C\left(\frac{\sqrt{2} \left(2 x + 1\right)}{2 \sqrt{\pi}}\right) + \sin{\left(\frac{1}{4} \right)} S\left(\frac{\sqrt{2} \left(2 x + 1\right)}{2 \sqrt{\pi}}\right)\right)}{4}
Simplificar
Respuesta [src] 
               /          /   ___  \    /   ___  \         \                /          /    ___ \    /    ___ \         \
    ___   ____ |          | \/ 2   |    | \/ 2   |         |     ___   ____ |          |3*\/ 2  |    |3*\/ 2  |         |
  \/ 2 *\/ pi *|cos(1/4)*C|--------| + S|--------|*sin(1/4)|   \/ 2 *\/ pi *|cos(1/4)*C|--------| + S|--------|*sin(1/4)|
               |          |    ____|    |    ____|         |                |          |    ____|    |    ____|         |
               \          \2*\/ pi /    \2*\/ pi /         /                \          \2*\/ pi /    \2*\/ pi /         /
- ---------------------------------------------------------- + ----------------------------------------------------------
                              4                                                            4
−2π(sin⁡(14)S(22π)+cos⁡(14)C(22π))4+2π(sin⁡(14)S(322π)+cos⁡(14)C(322π))4- \frac{\sqrt{2} \sqrt{\pi} \left(\sin{\left(\frac{1}{4} \right)} S\left(\frac{\sqrt{2}}{2 \sqrt{\pi}}\right) + \cos{\left(\frac{1}{4} \right)} C\left(\frac{\sqrt{2}}{2 \sqrt{\pi}}\right)\right)}{4} + \frac{\sqrt{2} \sqrt{\pi} \left(\sin{\left(\frac{1}{4} \right)} S\left(\frac{3 \sqrt{2}}{2 \sqrt{\pi}}\right) + \cos{\left(\frac{1}{4} \right)} C\left(\frac{3 \sqrt{2}}{2 \sqrt{\pi}}\right)\right)}{4} 
=
= 
               /          /   ___  \    /   ___  \         \                /          /    ___ \    /    ___ \         \
    ___   ____ |          | \/ 2   |    | \/ 2   |         |     ___   ____ |          |3*\/ 2  |    |3*\/ 2  |         |
  \/ 2 *\/ pi *|cos(1/4)*C|--------| + S|--------|*sin(1/4)|   \/ 2 *\/ pi *|cos(1/4)*C|--------| + S|--------|*sin(1/4)|
               |          |    ____|    |    ____|         |                |          |    ____|    |    ____|         |
               \          \2*\/ pi /    \2*\/ pi /         /                \          \2*\/ pi /    \2*\/ pi /         /
- ---------------------------------------------------------- + ----------------------------------------------------------
                              4                                                            4
−2π(sin⁡(14)S(22π)+cos⁡(14)C(22π))4+2π(sin⁡(14)S(322π)+cos⁡(14)C(322π))4- \frac{\sqrt{2} \sqrt{\pi} \left(\sin{\left(\frac{1}{4} \right)} S\left(\frac{\sqrt{2}}{2 \sqrt{\pi}}\right) + \cos{\left(\frac{1}{4} \right)} C\left(\frac{\sqrt{2}}{2 \sqrt{\pi}}\right)\right)}{4} + \frac{\sqrt{2} \sqrt{\pi} \left(\sin{\left(\frac{1}{4} \right)} S\left(\frac{3 \sqrt{2}}{2 \sqrt{\pi}}\right) + \cos{\left(\frac{1}{4} \right)} C\left(\frac{3 \sqrt{2}}{2 \sqrt{\pi}}\right)\right)}{4} 
-sqrt(2)*sqrt(pi)*(cos(1/4)*fresnelc(sqrt(2)/(2*sqrt(pi))) + fresnels(sqrt(2)/(2*sqrt(pi)))*sin(1/4))/4 + sqrt(2)*sqrt(pi)*(cos(1/4)*fresnelc(3*sqrt(2)/(2*sqrt(pi))) + fresnels(3*sqrt(2)/(2*sqrt(pi)))*sin(1/4))/4

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

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