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Resolución de integrales/ 1/x^2/ x^(2*a)*sin(1/x^2)

Integral de x^(2*a)*sin(1/x^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 oo                
  /                
 |                 
 |   2*a    /1 \   
 |  x   *sin|--| dx
 |          | 2|   
 |          \x /   
 |                 
/                  
1
∫1∞x2asin⁡(1x2) dx\int\limits_{1}^{\infty} x^{2 a} \sin{\left(\frac{1}{x^{2}} \right)}\, dx 
Integral(x^(2*a)*sin(1/(x^2)), (x, 1, oo))
Respuesta (Indefinida) [src] 
                                                                   
                                                /  1   a    |     \
                                             _  |  - - -    | -1  |
                          2*a      /1   a\  |_  |  4   2    | ----|
                         x   *Gamma|- - -|* |   |           |    4|
  /                                \4   2/ 1  2 |     5   a | 4*x |
 |                                              |3/2, - - - |     |
 |  2*a    /1 \                                 \     4   2 |     /
 | x   *sin|--| dx = C - ------------------------------------------
 |         | 2|                                /5   a\             
 |         \x /                       4*x*Gamma|- - -|             
 |                                             \4   2/             
/
∫x2asin⁡(1x2) dx=C−x2aΓ(14−a2)1F2(14−a232,54−a2|−14x4)4xΓ(54−a2)\int x^{2 a} \sin{\left(\frac{1}{x^{2}} \right)}\, dx = C - \frac{x^{2 a} \Gamma\left(\frac{1}{4} - \frac{a}{2}\right) {{}_{1}F_{2}\left(\begin{matrix} \frac{1}{4} - \frac{a}{2} \\ \frac{3}{2}, \frac{5}{4} - \frac{a}{2} \end{matrix}\middle| {- \frac{1}{4 x^{4}}} \right)}}{4 x \Gamma\left(\frac{5}{4} - \frac{a}{2}\right)}
Simplificar
Respuesta [src] 
/                                                                                                          
|                  /  1   a    |     \                                                                     
|               _  |  - - -    |     |                                                                     
|     /1   a\  |_  |  4   2    |     |                                                                     
|Gamma|- - -|* |   |           | -1/4|                                                                     
|     \4   2/ 1  2 |     5   a |     |                                                                     
|                  |3/2, - - - |     |                                                                     
|                  \     4   2 |     /         /    re(a)      5   re(a)      3   re(a)      7   re(a)    \
|-------------------------------------  for And|1 + ----- < 2, - + ----- < 2, - + ----- < 2, - + ----- < 2|
|                   /5   a\                    \      2        4     2        2     2        4     2      /
|            4*Gamma|- - -|                                                                                
<                   \4   2/                                                                                
|                                                                                                          
|          oo                                                                                              
|           /                                                                                              
|          |                                                                                               
|          |   2*a    /1 \                                                                                 
|          |  x   *sin|--| dx                                        otherwise                             
|          |          | 2|                                                                                 
|          |          \x /                                                                                 
|          |                                                                                               
|         /                                                                                                
\         1
{Γ(14−a2)1F2(14−a232,54−a2|−14)4Γ(54−a2)for re⁡(a)2+1<2∧re⁡(a)2+54<2∧re⁡(a)2+32<2∧re⁡(a)2+74<2∫1∞x2asin⁡(1x2) dxotherwise\begin{cases} \frac{\Gamma\left(\frac{1}{4} - \frac{a}{2}\right) {{}_{1}F_{2}\left(\begin{matrix} \frac{1}{4} - \frac{a}{2} \\ \frac{3}{2}, \frac{5}{4} - \frac{a}{2} \end{matrix}\middle| {- \frac{1}{4}} \right)}}{4 \Gamma\left(\frac{5}{4} - \frac{a}{2}\right)} & \text{for}\: \frac{\operatorname{re}{\left(a\right)}}{2} + 1 < 2 \wedge \frac{\operatorname{re}{\left(a\right)}}{2} + \frac{5}{4} < 2 \wedge \frac{\operatorname{re}{\left(a\right)}}{2} + \frac{3}{2} < 2 \wedge \frac{\operatorname{re}{\left(a\right)}}{2} + \frac{7}{4} < 2 \\\int\limits_{1}^{\infty} x^{2 a} \sin{\left(\frac{1}{x^{2}} \right)}\, dx & \text{otherwise} \end{cases} 
=
= 
/                                                                                                          
|                  /  1   a    |     \                                                                     
|               _  |  - - -    |     |                                                                     
|     /1   a\  |_  |  4   2    |     |                                                                     
|Gamma|- - -|* |   |           | -1/4|                                                                     
|     \4   2/ 1  2 |     5   a |     |                                                                     
|                  |3/2, - - - |     |                                                                     
|                  \     4   2 |     /         /    re(a)      5   re(a)      3   re(a)      7   re(a)    \
|-------------------------------------  for And|1 + ----- < 2, - + ----- < 2, - + ----- < 2, - + ----- < 2|
|                   /5   a\                    \      2        4     2        2     2        4     2      /
|            4*Gamma|- - -|                                                                                
<                   \4   2/                                                                                
|                                                                                                          
|          oo                                                                                              
|           /                                                                                              
|          |                                                                                               
|          |   2*a    /1 \                                                                                 
|          |  x   *sin|--| dx                                        otherwise                             
|          |          | 2|                                                                                 
|          |          \x /                                                                                 
|          |                                                                                               
|         /                                                                                                
\         1
{Γ(14−a2)1F2(14−a232,54−a2|−14)4Γ(54−a2)for re⁡(a)2+1<2∧re⁡(a)2+54<2∧re⁡(a)2+32<2∧re⁡(a)2+74<2∫1∞x2asin⁡(1x2) dxotherwise\begin{cases} \frac{\Gamma\left(\frac{1}{4} - \frac{a}{2}\right) {{}_{1}F_{2}\left(\begin{matrix} \frac{1}{4} - \frac{a}{2} \\ \frac{3}{2}, \frac{5}{4} - \frac{a}{2} \end{matrix}\middle| {- \frac{1}{4}} \right)}}{4 \Gamma\left(\frac{5}{4} - \frac{a}{2}\right)} & \text{for}\: \frac{\operatorname{re}{\left(a\right)}}{2} + 1 < 2 \wedge \frac{\operatorname{re}{\left(a\right)}}{2} + \frac{5}{4} < 2 \wedge \frac{\operatorname{re}{\left(a\right)}}{2} + \frac{3}{2} < 2 \wedge \frac{\operatorname{re}{\left(a\right)}}{2} + \frac{7}{4} < 2 \\\int\limits_{1}^{\infty} x^{2 a} \sin{\left(\frac{1}{x^{2}} \right)}\, dx & \text{otherwise} \end{cases} 
Piecewise((gamma(1/4 - a/2)*hyper((1/4 - a/2,), (3/2, 5/4 - a/2), -1/4)/(4*gamma(5/4 - a/2)), (1 + re(a)/2 < 2)∧(5/4 + re(a)/2 < 2)∧(3/2 + re(a)/2 < 2)∧(7/4 + re(a)/2 < 2)), (Integral(x^(2*a)*sin(x^(-2)), (x, 1, oo)), True))

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

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