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

Integral de x*exp(-a*x^2)*cos(b*x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 oo                     
  /                     
 |                      
 |         2            
 |     -a*x             
 |  x*e     *cos(b*x) dx
 |                      
/                       
0
∫0∞xe−ax2cos⁡(bx) dx\int\limits_{0}^{\infty} x e^{- a x^{2}} \cos{\left(b x \right)}\, dx 
Integral((x*exp((-a)*x^2))*cos(b*x), (x, 0, oo))
Respuesta [src] 
/       /                                   2\                                                                 
|       |    ____    /  I*b  \             b |    2                                                            
|       |  \/ pi *erf|-------|            ---|  -b                                                             
|       |            |    ___|       ___  4*a|  ----                                                           
|       |            \2*\/ a /   I*\/ a *e   |  4*a                                                            
|-2*I*b*|- ------------------- + ------------|*e                                                               
|       \           8                4*b     /              /   /                           pi\             pi\
|---------------------------------------------------  for Or|And|2*|arg(b)| = 0, |arg(a)| < --|, |arg(a)| < --|
|                         3/2                               \   \                           2 /             2 /
|                        a                                                                                     
<                                                                                                              
|              oo                                                                                              
|               /                                                                                              
|              |                                                                                               
|              |                  2                                                                            
|              |              -a*x                                                                             
|              |  x*cos(b*x)*e      dx                                        otherwise                        
|              |                                                                                               
|             /                                                                                                
|             0                                                                                                
\
{−2ib(iaeb24a4b−πerf⁡(ib2a)8)e−b24aa32for (2∣arg⁡(b)∣=0∧∣arg⁡(a)∣<π2)∨∣arg⁡(a)∣<π2∫0∞xe−ax2cos⁡(bx) dxotherwise\begin{cases} - \frac{2 i b \left(\frac{i \sqrt{a} e^{\frac{b^{2}}{4 a}}}{4 b} - \frac{\sqrt{\pi} \operatorname{erf}{\left(\frac{i b}{2 \sqrt{a}} \right)}}{8}\right) e^{- \frac{b^{2}}{4 a}}}{a^{\frac{3}{2}}} & \text{for}\: \left(2 \left|{\arg{\left(b \right)}}\right| = 0 \wedge \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2}\right) \vee \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2} \\\int\limits_{0}^{\infty} x e^{- a x^{2}} \cos{\left(b x \right)}\, dx & \text{otherwise} \end{cases} 
=
= 
/       /                                   2\                                                                 
|       |    ____    /  I*b  \             b |    2                                                            
|       |  \/ pi *erf|-------|            ---|  -b                                                             
|       |            |    ___|       ___  4*a|  ----                                                           
|       |            \2*\/ a /   I*\/ a *e   |  4*a                                                            
|-2*I*b*|- ------------------- + ------------|*e                                                               
|       \           8                4*b     /              /   /                           pi\             pi\
|---------------------------------------------------  for Or|And|2*|arg(b)| = 0, |arg(a)| < --|, |arg(a)| < --|
|                         3/2                               \   \                           2 /             2 /
|                        a                                                                                     
<                                                                                                              
|              oo                                                                                              
|               /                                                                                              
|              |                                                                                               
|              |                  2                                                                            
|              |              -a*x                                                                             
|              |  x*cos(b*x)*e      dx                                        otherwise                        
|              |                                                                                               
|             /                                                                                                
|             0                                                                                                
\
{−2ib(iaeb24a4b−πerf⁡(ib2a)8)e−b24aa32for (2∣arg⁡(b)∣=0∧∣arg⁡(a)∣<π2)∨∣arg⁡(a)∣<π2∫0∞xe−ax2cos⁡(bx) dxotherwise\begin{cases} - \frac{2 i b \left(\frac{i \sqrt{a} e^{\frac{b^{2}}{4 a}}}{4 b} - \frac{\sqrt{\pi} \operatorname{erf}{\left(\frac{i b}{2 \sqrt{a}} \right)}}{8}\right) e^{- \frac{b^{2}}{4 a}}}{a^{\frac{3}{2}}} & \text{for}\: \left(2 \left|{\arg{\left(b \right)}}\right| = 0 \wedge \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2}\right) \vee \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2} \\\int\limits_{0}^{\infty} x e^{- a x^{2}} \cos{\left(b x \right)}\, dx & \text{otherwise} \end{cases} 
Piecewise((-2*i*b*(-sqrt(pi)*erf(i*b/(2*sqrt(a)))/8 + i*sqrt(a)*exp(b^2/(4*a))/(4*b))*exp(-b^2/(4*a))/a^(3/2), (Abs(arg(a)) < pi/2)∨((2*Abs(arg(b)) = 0))∧(Abs(arg(a)) < pi/2)), (Integral(x*cos(b*x)*exp(-a*x^2), (x, 0, oo)), True))

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

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