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Resolución de integrales/ cos(x*a)/((x^2+b^2)^2)

Integral de cos(x*a)/((x^2+b^2)^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 oo              
  /              
 |               
 |   cos(x*a)    
 |  ---------- dx
 |           2   
 |  / 2    2\    
 |  \x  + b /    
 |               
/                
0
∫0∞cos⁡(ax)(b2+x2)2 dx\int\limits_{0}^{\infty} \frac{\cos{\left(a x \right)}}{\left(b^{2} + x^{2}\right)^{2}}\, dx 
Integral(cos(x*a)/(x^2 + b^2)^2, (x, 0, oo))
Respuesta (Indefinida) [src] 
  /                      /             
 |                      |              
 |  cos(x*a)            |  cos(x*a)    
 | ---------- dx = C +  | ---------- dx
 |          2           |          2   
 | / 2    2\            | / 2    2\    
 | \x  + b /            | \b  + x /    
 |                      |              
/                      /
∫cos⁡(ax)(b2+x2)2 dx=C+∫cos⁡(ax)(b2+x2)2 dx\int \frac{\cos{\left(a x \right)}}{\left(b^{2} + x^{2}\right)^{2}}\, dx = C + \int \frac{\cos{\left(a x \right)}}{\left(b^{2} + x^{2}\right)^{2}}\, dx
Simplificar
Respuesta [src] 
/       /                                               ____                             \                                          
|  ____ |  ____  4  4 /  sinh(a*b)   cosh(a*b)\   a*b*\/ pi *(-a*b*sinh(a*b) + cosh(a*b))|                                          
|\/ pi *|\/ pi *a *b *|- --------- + ---------| + ---------------------------------------|                                          
|       |             |      3  3        2  2 |                      4                   |                                          
|       \             \   4*a *b      4*a *b  /                                          /                                          
|-----------------------------------------------------------------------------------------  for And(2*|arg(a)| = 0, 2*|arg(b)| < pi)
|                                              4                                                                                    
|                                           a*b                                                                                     
|                                                                                                                                   
|                                     oo                                                                                            
<                                      /                                                                                            
|                                     |                                                                                             
|                                     |   cos(a*x)                                                                                  
|                                     |  ---------- dx                                                     otherwise                
|                                     |           2                                                                                 
|                                     |  / 2    2\                                                                                  
|                                     |  \b  + x /                                                                                  
|                                     |                                                                                             
|                                    /                                                                                              
|                                    0                                                                                              
\
{π(πa4b4(cosh⁡(ab)4a2b2−sinh⁡(ab)4a3b3)+πab(−absinh⁡(ab)+cosh⁡(ab))4)ab4for 2∣arg⁡(a)∣=0∧2∣arg⁡(b)∣<π∫0∞cos⁡(ax)(b2+x2)2 dxotherwise\begin{cases} \frac{\sqrt{\pi} \left(\sqrt{\pi} a^{4} b^{4} \left(\frac{\cosh{\left(a b \right)}}{4 a^{2} b^{2}} - \frac{\sinh{\left(a b \right)}}{4 a^{3} b^{3}}\right) + \frac{\sqrt{\pi} a b \left(- a b \sinh{\left(a b \right)} + \cosh{\left(a b \right)}\right)}{4}\right)}{a b^{4}} & \text{for}\: 2 \left|{\arg{\left(a \right)}}\right| = 0 \wedge 2 \left|{\arg{\left(b \right)}}\right| < \pi \\\int\limits_{0}^{\infty} \frac{\cos{\left(a x \right)}}{\left(b^{2} + x^{2}\right)^{2}}\, dx & \text{otherwise} \end{cases} 
=
= 
/       /                                               ____                             \                                          
|  ____ |  ____  4  4 /  sinh(a*b)   cosh(a*b)\   a*b*\/ pi *(-a*b*sinh(a*b) + cosh(a*b))|                                          
|\/ pi *|\/ pi *a *b *|- --------- + ---------| + ---------------------------------------|                                          
|       |             |      3  3        2  2 |                      4                   |                                          
|       \             \   4*a *b      4*a *b  /                                          /                                          
|-----------------------------------------------------------------------------------------  for And(2*|arg(a)| = 0, 2*|arg(b)| < pi)
|                                              4                                                                                    
|                                           a*b                                                                                     
|                                                                                                                                   
|                                     oo                                                                                            
<                                      /                                                                                            
|                                     |                                                                                             
|                                     |   cos(a*x)                                                                                  
|                                     |  ---------- dx                                                     otherwise                
|                                     |           2                                                                                 
|                                     |  / 2    2\                                                                                  
|                                     |  \b  + x /                                                                                  
|                                     |                                                                                             
|                                    /                                                                                              
|                                    0                                                                                              
\
{π(πa4b4(cosh⁡(ab)4a2b2−sinh⁡(ab)4a3b3)+πab(−absinh⁡(ab)+cosh⁡(ab))4)ab4for 2∣arg⁡(a)∣=0∧2∣arg⁡(b)∣<π∫0∞cos⁡(ax)(b2+x2)2 dxotherwise\begin{cases} \frac{\sqrt{\pi} \left(\sqrt{\pi} a^{4} b^{4} \left(\frac{\cosh{\left(a b \right)}}{4 a^{2} b^{2}} - \frac{\sinh{\left(a b \right)}}{4 a^{3} b^{3}}\right) + \frac{\sqrt{\pi} a b \left(- a b \sinh{\left(a b \right)} + \cosh{\left(a b \right)}\right)}{4}\right)}{a b^{4}} & \text{for}\: 2 \left|{\arg{\left(a \right)}}\right| = 0 \wedge 2 \left|{\arg{\left(b \right)}}\right| < \pi \\\int\limits_{0}^{\infty} \frac{\cos{\left(a x \right)}}{\left(b^{2} + x^{2}\right)^{2}}\, dx & \text{otherwise} \end{cases} 
Piecewise((sqrt(pi)*(sqrt(pi)*a^4*b^4*(-sinh(a*b)/(4*a^3*b^3) + cosh(a*b)/(4*a^2*b^2)) + a*b*sqrt(pi)*(-a*b*sinh(a*b) + cosh(a*b))/4)/(a*b^4), (2*Abs(arg(a)) = 0))∧(2*Abs(arg(b)) < pi), (Integral(cos(a*x)/(b^2 + x^2)^2, (x, 0, oo)), True))

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

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