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Resolución de integrales/ sinx/(x)^(a)

Integral de sinx/(x)^(a) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 oo          
  /          
 |           
 |  sin(x)   
 |  ------ dx
 |     a     
 |    x      
 |           
/            
1
∫1∞sin⁡(x)xa dx\int\limits_{1}^{\infty} \frac{\sin{\left(x \right)}}{x^{a}}\, dx 
Integral(sin(x)/x^a, (x, 1, oo))
Respuesta (Indefinida) [src] 
  /                  /             
 |                  |              
 | sin(x)           |  -a          
 | ------ dx = C +  | x  *sin(x) dx
 |    a             |              
 |   x             /               
 |                                 
/
∫sin⁡(x)xa dx=C+∫x−asin⁡(x) dx\int \frac{\sin{\left(x \right)}}{x^{a}}\, dx = C + \int x^{- a} \sin{\left(x \right)}\, dx
Simplificar
Respuesta [src] 
/       /                                                            \                                                   
|       |                                         /      a    |     \|                                                   
|       |                                      _  |  1 - -    |     ||                                                   
|       |                           /     a\  |_  |      2    |     ||                                                   
|       |                      Gamma|-1 + -|* |   |           | -1/4||                                                   
|       | 1 - a      /    a\        \     2/ 1  2 |         a |     ||                                                   
|       |2     *Gamma|1 - -|                      |3/2, 2 - - |     ||                                                   
|  ____ |            \    2/                      \         2 |     /|                                                   
|\/ pi *|------------------- + --------------------------------------|                                                   
|       |         /1   a\                   ____      /a\            |                                                   
|       |    Gamma|- + -|                 \/ pi *Gamma|-|            |                                                   
<       \         \2   2/                             \2/            /                                                   
|---------------------------------------------------------------------  for And(-3/2 + re(a) > -3/2, -1/2 + re(a) > -3/2)
|                                  2                                                                                     
|                                                                                                                        
|                           oo                                                                                           
|                            /                                                                                           
|                           |                                                                                            
|                           |   -a                                                                                       
|                           |  x  *sin(x) dx                                                otherwise                    
|                           |                                                                                            
|                          /                                                                                             
\                          1
{π(21−aΓ(1−a2)Γ(a2+12)+Γ(a2−1)1F2(1−a232,2−a2|−14)πΓ(a2))2for re⁡(a)−32>−32∧re⁡(a)−12>−32∫1∞x−asin⁡(x) dxotherwise\begin{cases} \frac{\sqrt{\pi} \left(\frac{2^{1 - a} \Gamma\left(1 - \frac{a}{2}\right)}{\Gamma\left(\frac{a}{2} + \frac{1}{2}\right)} + \frac{\Gamma\left(\frac{a}{2} - 1\right) {{}_{1}F_{2}\left(\begin{matrix} 1 - \frac{a}{2} \\ \frac{3}{2}, 2 - \frac{a}{2} \end{matrix}\middle| {- \frac{1}{4}} \right)}}{\sqrt{\pi} \Gamma\left(\frac{a}{2}\right)}\right)}{2} & \text{for}\: \operatorname{re}{\left(a\right)} - \frac{3}{2} > - \frac{3}{2} \wedge \operatorname{re}{\left(a\right)} - \frac{1}{2} > - \frac{3}{2} \\\int\limits_{1}^{\infty} x^{- a} \sin{\left(x \right)}\, dx & \text{otherwise} \end{cases} 
=
= 
/       /                                                            \                                                   
|       |                                         /      a    |     \|                                                   
|       |                                      _  |  1 - -    |     ||                                                   
|       |                           /     a\  |_  |      2    |     ||                                                   
|       |                      Gamma|-1 + -|* |   |           | -1/4||                                                   
|       | 1 - a      /    a\        \     2/ 1  2 |         a |     ||                                                   
|       |2     *Gamma|1 - -|                      |3/2, 2 - - |     ||                                                   
|  ____ |            \    2/                      \         2 |     /|                                                   
|\/ pi *|------------------- + --------------------------------------|                                                   
|       |         /1   a\                   ____      /a\            |                                                   
|       |    Gamma|- + -|                 \/ pi *Gamma|-|            |                                                   
<       \         \2   2/                             \2/            /                                                   
|---------------------------------------------------------------------  for And(-3/2 + re(a) > -3/2, -1/2 + re(a) > -3/2)
|                                  2                                                                                     
|                                                                                                                        
|                           oo                                                                                           
|                            /                                                                                           
|                           |                                                                                            
|                           |   -a                                                                                       
|                           |  x  *sin(x) dx                                                otherwise                    
|                           |                                                                                            
|                          /                                                                                             
\                          1
{π(21−aΓ(1−a2)Γ(a2+12)+Γ(a2−1)1F2(1−a232,2−a2|−14)πΓ(a2))2for re⁡(a)−32>−32∧re⁡(a)−12>−32∫1∞x−asin⁡(x) dxotherwise\begin{cases} \frac{\sqrt{\pi} \left(\frac{2^{1 - a} \Gamma\left(1 - \frac{a}{2}\right)}{\Gamma\left(\frac{a}{2} + \frac{1}{2}\right)} + \frac{\Gamma\left(\frac{a}{2} - 1\right) {{}_{1}F_{2}\left(\begin{matrix} 1 - \frac{a}{2} \\ \frac{3}{2}, 2 - \frac{a}{2} \end{matrix}\middle| {- \frac{1}{4}} \right)}}{\sqrt{\pi} \Gamma\left(\frac{a}{2}\right)}\right)}{2} & \text{for}\: \operatorname{re}{\left(a\right)} - \frac{3}{2} > - \frac{3}{2} \wedge \operatorname{re}{\left(a\right)} - \frac{1}{2} > - \frac{3}{2} \\\int\limits_{1}^{\infty} x^{- a} \sin{\left(x \right)}\, dx & \text{otherwise} \end{cases} 
Piecewise((sqrt(pi)*(2^(1 - a)*gamma(1 - a/2)/gamma(1/2 + a/2) + gamma(-1 + a/2)*hyper((1 - a/2,), (3/2, 2 - a/2), -1/4)/(sqrt(pi)*gamma(a/2)))/2, (-3/2 + re(a) > -3/2)∧(-1/2 + re(a) > -3/2)), (Integral(x^(-a)*sin(x), (x, 1, oo)), True))

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

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