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Resolución de integrales/ x^(3)/ x^(2)/ (x^(3))*(sin(4*(x^(2))-6))

Integral de (x^(3))(sin(4(x^(2))-6)) dx

Solución

Ha introducido [src]

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0

$$\int\limits_{0}^{1} x^{3} \sin{\left(4 x^{2} - 6 \right)}\, dx$$

Integral(x^3*sin(4*x^2 - 6), (x, 0, 1))

Solución detallada

Usamos la integración por partes:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

que $u{\left(x \right)} = x^{3}$ y que $\operatorname{dv}{\left(x \right)} = \sin{\left(4 x^{2} - 6 \right)}$ .

Entonces $\operatorname{du}{\left(x \right)} = 3 x^{2}$ .

Para buscar $v{\left(x \right)}$ :
Ahora resolvemos podintegral.
La integral del producto de una función por una constante es la constante por la integral de esta función:

$\int \frac{3 \sqrt{2} \sqrt{\pi} x^{2} \left(- \sin{\left(6 \right)} C\left(\frac{2 \sqrt{2} x}{\sqrt{\pi}}\right) + \cos{\left(6 \right)} S\left(\frac{2 \sqrt{2} x}{\sqrt{\pi}}\right)\right)}{4}\, dx = \frac{3 \sqrt{2} \sqrt{\pi} \int x^{2} \left(- \sin{\left(6 \right)} C\left(\frac{2 \sqrt{2} x}{\sqrt{\pi}}\right) + \cos{\left(6 \right)} S\left(\frac{2 \sqrt{2} x}{\sqrt{\pi}}\right)\right)\, dx}{4}$
1. Vuelva a escribir el integrando:
  
  $x^{2} \left(- \sin{\left(6 \right)} C\left(\frac{2 \sqrt{2} x}{\sqrt{\pi}}\right) + \cos{\left(6 \right)} S\left(\frac{2 \sqrt{2} x}{\sqrt{\pi}}\right)\right) = - x^{2} \sin{\left(6 \right)} C\left(\frac{2 \sqrt{2} x}{\sqrt{\pi}}\right) + x^{2} \cos{\left(6 \right)} S\left(\frac{2 \sqrt{2} x}{\sqrt{\pi}}\right)$
2. Integramos término a término:
  1. La integral del producto de una función por una constante es la constante por la integral de esta función:
    
    $\int \left(- x^{2} \sin{\left(6 \right)} C\left(\frac{2 \sqrt{2} x}{\sqrt{\pi}}\right)\right)\, dx = - \sin{\left(6 \right)} \int x^{2} C\left(\frac{2 \sqrt{2} x}{\sqrt{\pi}}\right)\, dx$
    1. No puedo encontrar los pasos en la búsqueda de esta integral.
      
      Pero la integral
      
      $\frac{x^{3} C\left(\frac{2 \sqrt{2} x}{\sqrt{\pi}}\right) \Gamma\left(\frac{1}{4}\right)}{12 \Gamma\left(\frac{5}{4}\right)} - \frac{\sqrt{2} x^{2} \sin{\left(4 x^{2} \right)} \Gamma\left(\frac{1}{4}\right)}{48 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right)} - \frac{\sqrt{2} \cos{\left(4 x^{2} \right)} \Gamma\left(\frac{1}{4}\right)}{192 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right)}$
    Por lo tanto, el resultado es: $- \left(\frac{x^{3} C\left(\frac{2 \sqrt{2} x}{\sqrt{\pi}}\right) \Gamma\left(\frac{1}{4}\right)}{12 \Gamma\left(\frac{5}{4}\right)} - \frac{\sqrt{2} x^{2} \sin{\left(4 x^{2} \right)} \Gamma\left(\frac{1}{4}\right)}{48 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right)} - \frac{\sqrt{2} \cos{\left(4 x^{2} \right)} \Gamma\left(\frac{1}{4}\right)}{192 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right)}\right) \sin{\left(6 \right)}$
  1. La integral del producto de una función por una constante es la constante por la integral de esta función:
    
    $\int x^{2} \cos{\left(6 \right)} S\left(\frac{2 \sqrt{2} x}{\sqrt{\pi}}\right)\, dx = \cos{\left(6 \right)} \int x^{2} S\left(\frac{2 \sqrt{2} x}{\sqrt{\pi}}\right)\, dx$
    1. No puedo encontrar los pasos en la búsqueda de esta integral.
      
      Pero la integral
      
      $\frac{x^{3} S\left(\frac{2 \sqrt{2} x}{\sqrt{\pi}}\right) \Gamma\left(\frac{3}{4}\right)}{4 \Gamma\left(\frac{7}{4}\right)} + \frac{\sqrt{2} x^{2} \cos{\left(4 x^{2} \right)} \Gamma\left(\frac{3}{4}\right)}{16 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right)} - \frac{\sqrt{2} \sin{\left(4 x^{2} \right)} \Gamma\left(\frac{3}{4}\right)}{64 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right)}$
    Por lo tanto, el resultado es: $\left(\frac{x^{3} S\left(\frac{2 \sqrt{2} x}{\sqrt{\pi}}\right) \Gamma\left(\frac{3}{4}\right)}{4 \Gamma\left(\frac{7}{4}\right)} + \frac{\sqrt{2} x^{2} \cos{\left(4 x^{2} \right)} \Gamma\left(\frac{3}{4}\right)}{16 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right)} - \frac{\sqrt{2} \sin{\left(4 x^{2} \right)} \Gamma\left(\frac{3}{4}\right)}{64 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right)}\right) \cos{\left(6 \right)}$
  El resultado es: $- \left(\frac{x^{3} C\left(\frac{2 \sqrt{2} x}{\sqrt{\pi}}\right) \Gamma\left(\frac{1}{4}\right)}{12 \Gamma\left(\frac{5}{4}\right)} - \frac{\sqrt{2} x^{2} \sin{\left(4 x^{2} \right)} \Gamma\left(\frac{1}{4}\right)}{48 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right)} - \frac{\sqrt{2} \cos{\left(4 x^{2} \right)} \Gamma\left(\frac{1}{4}\right)}{192 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right)}\right) \sin{\left(6 \right)} + \left(\frac{x^{3} S\left(\frac{2 \sqrt{2} x}{\sqrt{\pi}}\right) \Gamma\left(\frac{3}{4}\right)}{4 \Gamma\left(\frac{7}{4}\right)} + \frac{\sqrt{2} x^{2} \cos{\left(4 x^{2} \right)} \Gamma\left(\frac{3}{4}\right)}{16 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right)} - \frac{\sqrt{2} \sin{\left(4 x^{2} \right)} \Gamma\left(\frac{3}{4}\right)}{64 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right)}\right) \cos{\left(6 \right)}$
Por lo tanto, el resultado es: $\frac{3 \sqrt{2} \sqrt{\pi} \left(- \left(\frac{x^{3} C\left(\frac{2 \sqrt{2} x}{\sqrt{\pi}}\right) \Gamma\left(\frac{1}{4}\right)}{12 \Gamma\left(\frac{5}{4}\right)} - \frac{\sqrt{2} x^{2} \sin{\left(4 x^{2} \right)} \Gamma\left(\frac{1}{4}\right)}{48 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right)} - \frac{\sqrt{2} \cos{\left(4 x^{2} \right)} \Gamma\left(\frac{1}{4}\right)}{192 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right)}\right) \sin{\left(6 \right)} + \left(\frac{x^{3} S\left(\frac{2 \sqrt{2} x}{\sqrt{\pi}}\right) \Gamma\left(\frac{3}{4}\right)}{4 \Gamma\left(\frac{7}{4}\right)} + \frac{\sqrt{2} x^{2} \cos{\left(4 x^{2} \right)} \Gamma\left(\frac{3}{4}\right)}{16 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right)} - \frac{\sqrt{2} \sin{\left(4 x^{2} \right)} \Gamma\left(\frac{3}{4}\right)}{64 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right)}\right) \cos{\left(6 \right)}\right)}{4}$
Ahora simplificar:

$- \frac{\sqrt{2} \left(4 x^{2} \sin{\left(6 \right)} \sin{\left(4 x^{2} \right)} + 4 x^{2} \cos{\left(6 \right)} \cos{\left(4 x^{2} \right)} - \sin{\left(4 x^{2} \right)} \cos{\left(6 \right)} + \sin{\left(6 \right)} \cos{\left(4 x^{2} \right)}\right) \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right)}{64 \pi}$
Añadimos la constante de integración:

$- \frac{\sqrt{2} \left(4 x^{2} \sin{\left(6 \right)} \sin{\left(4 x^{2} \right)} + 4 x^{2} \cos{\left(6 \right)} \cos{\left(4 x^{2} \right)} - \sin{\left(4 x^{2} \right)} \cos{\left(6 \right)} + \sin{\left(6 \right)} \cos{\left(4 x^{2} \right)}\right) \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right)}{64 \pi}+ \mathrm{constant}$

Respuesta:

$- \frac{\sqrt{2} \left(4 x^{2} \sin{\left(6 \right)} \sin{\left(4 x^{2} \right)} + 4 x^{2} \cos{\left(6 \right)} \cos{\left(4 x^{2} \right)} - \sin{\left(4 x^{2} \right)} \cos{\left(6 \right)} + \sin{\left(6 \right)} \cos{\left(4 x^{2} \right)}\right) \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right)}{64 \pi}+ \mathrm{constant}$

Respuesta (Indefinida) [src]

                                            //    /      ___\                                                                        \          /    /      ___\                                                                        \       \                                                              
                                            || 3  |2*x*\/ 2 |                                                                        |          | 3  |2*x*\/ 2 |                                                                        |       |                                                              
                                            ||x *S|---------|*Gamma(3/4)                                                             |          |x *C|---------|*Gamma(1/4)                                                             |       |                                                              
                                            ||    |    ____ |                ___               /   2\     ___  2    /   2\           |          |    |    ____ |                ___    /   2\                ___  2               /   2\|       |                   /        /      ___\    /      ___\       \
                                 ___   ____ ||    \  \/ pi  /              \/ 2 *Gamma(3/4)*sin\4*x /   \/ 2 *x *cos\4*x /*Gamma(3/4)|          |    \  \/ pi  /              \/ 2 *cos\4*x /*Gamma(1/4)   \/ 2 *x *Gamma(1/4)*sin\4*x /|       |     ___   ____  3 |        |2*x*\/ 2 |    |2*x*\/ 2 |       |
  /                          3*\/ 2 *\/ pi *||-------------------------- - -------------------------- + -----------------------------|*cos(6) - |-------------------------- - -------------------------- - -----------------------------|*sin(6)|   \/ 2 *\/ pi *x *|cos(6)*S|---------| - C|---------|*sin(6)|
 |                                          ||       4*Gamma(7/4)                  ____                           ____               |          |      12*Gamma(5/4)                  ____                           ____               |       |                   |        |    ____ |    |    ____ |       |
 |  3    /   2    \                         \\                                64*\/ pi *Gamma(7/4)           16*\/ pi *Gamma(7/4)    /          \                               192*\/ pi *Gamma(5/4)           48*\/ pi *Gamma(5/4)    /       /                   \        \  \/ pi  /    \  \/ pi  /       /
 | x *sin\4*x  - 6/ dx = C - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + -----------------------------------------------------------
 |                                                                                                                                    4                                                                                                                                          4                             
/

$$\int x^{3} \sin{\left(4 x^{2} - 6 \right)}\, dx = C + \frac{\sqrt{2} \sqrt{\pi} x^{3} \left(- \sin{\left(6 \right)} C\left(\frac{2 \sqrt{2} x}{\sqrt{\pi}}\right) + \cos{\left(6 \right)} S\left(\frac{2 \sqrt{2} x}{\sqrt{\pi}}\right)\right)}{4} - \frac{3 \sqrt{2} \sqrt{\pi} \left(- \left(\frac{x^{3} C\left(\frac{2 \sqrt{2} x}{\sqrt{\pi}}\right) \Gamma\left(\frac{1}{4}\right)}{12 \Gamma\left(\frac{5}{4}\right)} - \frac{\sqrt{2} x^{2} \sin{\left(4 x^{2} \right)} \Gamma\left(\frac{1}{4}\right)}{48 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right)} - \frac{\sqrt{2} \cos{\left(4 x^{2} \right)} \Gamma\left(\frac{1}{4}\right)}{192 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right)}\right) \sin{\left(6 \right)} + \left(\frac{x^{3} S\left(\frac{2 \sqrt{2} x}{\sqrt{\pi}}\right) \Gamma\left(\frac{3}{4}\right)}{4 \Gamma\left(\frac{7}{4}\right)} + \frac{\sqrt{2} x^{2} \cos{\left(4 x^{2} \right)} \Gamma\left(\frac{3}{4}\right)}{16 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right)} - \frac{\sqrt{2} \sin{\left(4 x^{2} \right)} \Gamma\left(\frac{3}{4}\right)}{64 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right)}\right) \cos{\left(6 \right)}\right)}{4}$$

Simplificar

Gráfica

Trazar el gráfico f(x)

Respuesta [src]

  cos(2)   sin(2)   sin(6)
- ------ - ------ + ------
    8        32       32

$$- \frac{\sin{\left(2 \right)}}{32} + \frac{\sin{\left(6 \right)}}{32} - \frac{\cos{\left(2 \right)}}{8}$$

  cos(2)   sin(2)   sin(6)
- ------ - ------ + ------
    8        32       32

$$- \frac{\sin{\left(2 \right)}}{32} + \frac{\sin{\left(6 \right)}}{32} - \frac{\cos{\left(2 \right)}}{8}$$

-cos(2)/8 - sin(2)/32 + sin(6)/32

Respuesta numérica [src]

0.0148710756613738

0.0148710756613738

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

Otras calculadoras

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Expresiones idénticas

Expresiones semejantes

Expresiones con funciones

Integral de (x^(3))(sin(4(x^(2))-6)) dx

Límites de integración:

Gráfico:

Definida a trozos:

Solución

Otras calculadoras

¿Cómo usar?

Expresiones idénticas

Expresiones semejantes

Expresiones con funciones

Integral de (x^(3))*(sin(4*(x^(2))-6)) dx

Límites de integración:

Gráfico:

Definida a trozos:

Solución

Integral de (x^(3))(sin(4(x^(2))-6)) dx