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Resolución de integrales/ 1+x^2/ cos(k*x)/(1+x^2)

Integral de cos(k*x)/(1+x^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 oo            
  /            
 |             
 |  cos(k*x)   
 |  -------- dx
 |        2    
 |   1 + x     
 |             
/              
-oo
∫−∞∞cos⁡(kx)x2+1 dx\int\limits_{-\infty}^{\infty} \frac{\cos{\left(k x \right)}}{x^{2} + 1}\, dx 
Integral(cos(k*x)/(1 + x^2), (x, -oo, oo))
Respuesta (Indefinida) [src] 
  /                    /           
 |                    |            
 | cos(k*x)           | cos(k*x)   
 | -------- dx = C +  | -------- dx
 |       2            |       2    
 |  1 + x             |  1 + x     
 |                    |            
/                    /
∫cos⁡(kx)x2+1 dx=C+∫cos⁡(kx)x2+1 dx\int \frac{\cos{\left(k x \right)}}{x^{2} + 1}\, dx = C + \int \frac{\cos{\left(k x \right)}}{x^{2} + 1}\, dx
Simplificar
Respuesta [src] 
/  ____ /  ____             ____        \                    
|\/ pi *\\/ pi *cosh(k) - \/ pi *sinh(k)/  for 2*|arg(k)| = 0
|                                                            
|             oo                                             
|              /                                             
|             |                                              
<             |  cos(k*x)                                    
|             |  -------- dx                   otherwise     
|             |        2                                     
|             |   1 + x                                      
|             |                                              
|            /                                               
\            -oo
{π(−πsinh⁡(k)+πcosh⁡(k))for 2∣arg⁡(k)∣=0∫−∞∞cos⁡(kx)x2+1 dxotherwise\begin{cases} \sqrt{\pi} \left(- \sqrt{\pi} \sinh{\left(k \right)} + \sqrt{\pi} \cosh{\left(k \right)}\right) & \text{for}\: 2 \left|{\arg{\left(k \right)}}\right| = 0 \\\int\limits_{-\infty}^{\infty} \frac{\cos{\left(k x \right)}}{x^{2} + 1}\, dx & \text{otherwise} \end{cases} 
=
= 
/  ____ /  ____             ____        \                    
|\/ pi *\\/ pi *cosh(k) - \/ pi *sinh(k)/  for 2*|arg(k)| = 0
|                                                            
|             oo                                             
|              /                                             
|             |                                              
<             |  cos(k*x)                                    
|             |  -------- dx                   otherwise     
|             |        2                                     
|             |   1 + x                                      
|             |                                              
|            /                                               
\            -oo
{π(−πsinh⁡(k)+πcosh⁡(k))for 2∣arg⁡(k)∣=0∫−∞∞cos⁡(kx)x2+1 dxotherwise\begin{cases} \sqrt{\pi} \left(- \sqrt{\pi} \sinh{\left(k \right)} + \sqrt{\pi} \cosh{\left(k \right)}\right) & \text{for}\: 2 \left|{\arg{\left(k \right)}}\right| = 0 \\\int\limits_{-\infty}^{\infty} \frac{\cos{\left(k x \right)}}{x^{2} + 1}\, dx & \text{otherwise} \end{cases} 
Piecewise((sqrt(pi)*(sqrt(pi)*cosh(k) - sqrt(pi)*sinh(k)), 2*Abs(arg(k)) = 0), (Integral(cos(k*x)/(1 + x^2), (x, -oo, oo)), True))

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

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