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Resolución de integrales/ sin(x/4)/ √2xsin(x/4)

Integral de √2xsin(x/4) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 pi                  
  /                  
 |                   
 |    _____    /x\   
 |  \/ 2*x *sin|-| dx
 |             \4/   
 |                   
/                    
-pi
$$\int\limits_{- \pi}^{\pi} \sqrt{2 x} \sin{\left(\frac{x}{4} \right)}\, dx$$ 
Integral(sqrt(2*x)*sin(x/4), (x, -pi, pi))
Respuesta (Indefinida) [src] 
                              ___   ___                    
                            \/ 2 *\/ x                     
                                 /                         
  /                             |                          
 |                              |         ____      / 2\   
 |   _____    /x\               |        /  2       |u |   
 | \/ 2*x *sin|-| dx = C +      |      \/  u  *u*sin|--| du
 |            \4/               |                   \8 /   
 |                              |                          
/                              /
$$\int \sqrt{2 x} \sin{\left(\frac{x}{4} \right)}\, dx = C + \int\limits^{\sqrt{2} \sqrt{x}} u \sqrt{u^{2}} \sin{\left(\frac{u^{2}}{8} \right)}\, du$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
      /                                              /  ___\           \         /                                                  /  ___\           \
      |                                  ___   ____  |\/ 2 |           |         |                                      ___   ____  |\/ 2 |           |
      |      ___   ____              5*\/ 2 *\/ pi *C|-----|*Gamma(5/4)|         |        ___   ____              5*I*\/ 2 *\/ pi *C|-----|*Gamma(5/4)|
  ___ |  5*\/ 2 *\/ pi *Gamma(5/4)                   \  2  /           |     ___ |  5*I*\/ 2 *\/ pi *Gamma(5/4)                     \  2  /           |
\/ 2 *|- ------------------------- + ----------------------------------| - \/ 2 *|- --------------------------- + ------------------------------------|
      \         2*Gamma(9/4)                     Gamma(9/4)            /         \          2*Gamma(9/4)                       Gamma(9/4)             /
$$\sqrt{2} \left(- \frac{5 \sqrt{2} \sqrt{\pi} \Gamma\left(\frac{5}{4}\right)}{2 \Gamma\left(\frac{9}{4}\right)} + \frac{5 \sqrt{2} \sqrt{\pi} C\left(\frac{\sqrt{2}}{2}\right) \Gamma\left(\frac{5}{4}\right)}{\Gamma\left(\frac{9}{4}\right)}\right) - \sqrt{2} \left(- \frac{5 \sqrt{2} i \sqrt{\pi} \Gamma\left(\frac{5}{4}\right)}{2 \Gamma\left(\frac{9}{4}\right)} + \frac{5 \sqrt{2} i \sqrt{\pi} C\left(\frac{\sqrt{2}}{2}\right) \Gamma\left(\frac{5}{4}\right)}{\Gamma\left(\frac{9}{4}\right)}\right)$$ 
=
= 
      /                                              /  ___\           \         /                                                  /  ___\           \
      |                                  ___   ____  |\/ 2 |           |         |                                      ___   ____  |\/ 2 |           |
      |      ___   ____              5*\/ 2 *\/ pi *C|-----|*Gamma(5/4)|         |        ___   ____              5*I*\/ 2 *\/ pi *C|-----|*Gamma(5/4)|
  ___ |  5*\/ 2 *\/ pi *Gamma(5/4)                   \  2  /           |     ___ |  5*I*\/ 2 *\/ pi *Gamma(5/4)                     \  2  /           |
\/ 2 *|- ------------------------- + ----------------------------------| - \/ 2 *|- --------------------------- + ------------------------------------|
      \         2*Gamma(9/4)                     Gamma(9/4)            /         \          2*Gamma(9/4)                       Gamma(9/4)             /
$$\sqrt{2} \left(- \frac{5 \sqrt{2} \sqrt{\pi} \Gamma\left(\frac{5}{4}\right)}{2 \Gamma\left(\frac{9}{4}\right)} + \frac{5 \sqrt{2} \sqrt{\pi} C\left(\frac{\sqrt{2}}{2}\right) \Gamma\left(\frac{5}{4}\right)}{\Gamma\left(\frac{9}{4}\right)}\right) - \sqrt{2} \left(- \frac{5 \sqrt{2} i \sqrt{\pi} \Gamma\left(\frac{5}{4}\right)}{2 \Gamma\left(\frac{9}{4}\right)} + \frac{5 \sqrt{2} i \sqrt{\pi} C\left(\frac{\sqrt{2}}{2}\right) \Gamma\left(\frac{5}{4}\right)}{\Gamma\left(\frac{9}{4}\right)}\right)$$ 
sqrt(2)*(-5*sqrt(2)*sqrt(pi)*gamma(5/4)/(2*gamma(9/4)) + 5*sqrt(2)*sqrt(pi)*fresnelc(sqrt(2)/2)*gamma(5/4)/gamma(9/4)) - sqrt(2)*(-5*i*sqrt(2)*sqrt(pi)*gamma(5/4)/(2*gamma(9/4)) + 5*i*sqrt(2)*sqrt(pi)*fresnelc(sqrt(2)/2)*gamma(5/4)/gamma(9/4))
Respuesta numérica [src] 
(2.3356104029169 - 2.3356104029169j)
(2.3356104029169 - 2.3356104029169j)

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

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