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Resolución de integrales/ cosbx/ x^2dx/ e^(-ax^2)/ e^(-ax^2)*cosbx^2dx

Integral de e^(-ax^2)*cosbx^2dx dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 oo                    
  /                    
 |                     
 |       2             
 |   -a*x     2        
 |  E     *cos (b*x) dx
 |                     
/                      
0
∫0∞e−ax2cos⁡2(bx) dx\int\limits_{0}^{\infty} e^{- a x^{2}} \cos^{2}{\left(b x \right)}\, dx 
Integral(E^((-a)*x^2)*cos(b*x)^2, (x, 0, oo))
Respuesta [src] 
/                    2                                                                                  
|                  -b                                                                                   
|                  ----                                                                                 
|   ____     ____   a                                                                                   
| \/ pi    \/ pi *e              /            pi    /   /                           pi\             pi\\
|------- + ------------   for And||arg(a)| <= --, Or|And|2*|arg(b)| = 0, |arg(a)| < --|, |arg(a)| < --||
|    ___         ___             \            2     \   \                           2 /             2 //
|4*\/ a      4*\/ a                                                                                     
|                                                                                                       
< oo                                                                                                    
|  /                                                                                                    
| |                                                                                                     
| |                 2                                                                                   
| |     2       -a*x                                                                                    
| |  cos (b*x)*e      dx                                    otherwise                                   
| |                                                                                                     
|/                                                                                                      
|0                                                                                                      
\
{π4a+πe−b2a4afor ∣arg⁡(a)∣≤π2∧((2∣arg⁡(b)∣=0∧∣arg⁡(a)∣<π2)∨∣arg⁡(a)∣<π2)∫0∞e−ax2cos⁡2(bx) dxotherwise\begin{cases} \frac{\sqrt{\pi}}{4 \sqrt{a}} + \frac{\sqrt{\pi} e^{- \frac{b^{2}}{a}}}{4 \sqrt{a}} & \text{for}\: \left|{\arg{\left(a \right)}}\right| \leq \frac{\pi}{2} \wedge \left(\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}\right) \\\int\limits_{0}^{\infty} e^{- a x^{2}} \cos^{2}{\left(b x \right)}\, dx & \text{otherwise} \end{cases} 
=
= 
/                    2                                                                                  
|                  -b                                                                                   
|                  ----                                                                                 
|   ____     ____   a                                                                                   
| \/ pi    \/ pi *e              /            pi    /   /                           pi\             pi\\
|------- + ------------   for And||arg(a)| <= --, Or|And|2*|arg(b)| = 0, |arg(a)| < --|, |arg(a)| < --||
|    ___         ___             \            2     \   \                           2 /             2 //
|4*\/ a      4*\/ a                                                                                     
|                                                                                                       
< oo                                                                                                    
|  /                                                                                                    
| |                                                                                                     
| |                 2                                                                                   
| |     2       -a*x                                                                                    
| |  cos (b*x)*e      dx                                    otherwise                                   
| |                                                                                                     
|/                                                                                                      
|0                                                                                                      
\
{π4a+πe−b2a4afor ∣arg⁡(a)∣≤π2∧((2∣arg⁡(b)∣=0∧∣arg⁡(a)∣<π2)∨∣arg⁡(a)∣<π2)∫0∞e−ax2cos⁡2(bx) dxotherwise\begin{cases} \frac{\sqrt{\pi}}{4 \sqrt{a}} + \frac{\sqrt{\pi} e^{- \frac{b^{2}}{a}}}{4 \sqrt{a}} & \text{for}\: \left|{\arg{\left(a \right)}}\right| \leq \frac{\pi}{2} \wedge \left(\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}\right) \\\int\limits_{0}^{\infty} e^{- a x^{2}} \cos^{2}{\left(b x \right)}\, dx & \text{otherwise} \end{cases} 
Piecewise((sqrt(pi)/(4*sqrt(a)) + sqrt(pi)*exp(-b^2/a)/(4*sqrt(a)), (Abs(arg(a)) <= pi/2)∧((Abs(arg(a)) < pi/2)∨((2*Abs(arg(b)) = 0))∧(Abs(arg(a)) < pi/2))), (Integral(cos(b*x)^2*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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