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Resolución de integrales/ exp(-a*(cos(x))^2)+b*cos(x)*(1+erfc(b*cos(x)))

Integral de exp(-a*(cos(x))^2)+b*cos(x)*(1+erfc(b*cos(x))) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  pi                                                 
  --                                                 
  2                                                  
   /                                                 
  |                                                  
  |  /       2                                   \   
  |  | -a*cos (x)                                |   
  |  \e           + b*cos(x)*(1 + erfc(b*cos(x)))/ dx
  |                                                  
 /                                                   
-pi                                                  
----                                                 
 2
∫−π2π2(bcos⁡(x)(erfc⁡(bcos⁡(x))+1)+e−acos⁡2(x)) dx\int\limits_{- \frac{\pi}{2}}^{\frac{\pi}{2}} \left(b \cos{\left(x \right)} \left(\operatorname{erfc}{\left(b \cos{\left(x \right)} \right)} + 1\right) + e^{- a \cos^{2}{\left(x \right)}}\right)\, dx 
Integral(exp((-a)*cos(x)^2) + (b*cos(x))*(1 + erfc(b*cos(x))), (x, -pi/2, pi/2))
Respuesta (Indefinida) [src] 
  /                                                                                                    /              
 |                                                                                                    |               
 | /       2                                   \              /                                       |        2      
 | | -a*cos (x)                                |             |                                        |  -a*cos (x)   
 | \e           + b*cos(x)*(1 + erfc(b*cos(x)))/ dx = C + b* | cos(x)*erfc(b*cos(x)) dx + b*sin(x) +  | e           dx
 |                                                           |                                        |               
/                                                           /                                        /
∫(bcos⁡(x)(erfc⁡(bcos⁡(x))+1)+e−acos⁡2(x)) dx=C+bsin⁡(x)+b∫cos⁡(x)erfc⁡(bcos⁡(x)) dx+∫e−acos⁡2(x) dx\int \left(b \cos{\left(x \right)} \left(\operatorname{erfc}{\left(b \cos{\left(x \right)} \right)} + 1\right) + e^{- a \cos^{2}{\left(x \right)}}\right)\, dx = C + b \sin{\left(x \right)} + b \int \cos{\left(x \right)} \operatorname{erfc}{\left(b \cos{\left(x \right)} \right)}\, dx + \int e^{- a \cos^{2}{\left(x \right)}}\, dx
Simplificar
Respuesta [src] 
  pi                                                                              
  --                                                                              
  2                                                                               
   /                                                                              
  |                                                                               
  |  /                   2                                    2   \        2      
  |  |              a*cos (x)                            a*cos (x)|  -a*cos (x)   
  |  \1 + b*cos(x)*e          + b*cos(x)*erfc(b*cos(x))*e         /*e           dx
  |                                                                               
 /                                                                                
-pi                                                                               
----                                                                              
 2
∫−π2π2(beacos⁡2(x)cos⁡(x)erfc⁡(bcos⁡(x))+beacos⁡2(x)cos⁡(x)+1)e−acos⁡2(x) dx\int\limits_{- \frac{\pi}{2}}^{\frac{\pi}{2}} \left(b e^{a \cos^{2}{\left(x \right)}} \cos{\left(x \right)} \operatorname{erfc}{\left(b \cos{\left(x \right)} \right)} + b e^{a \cos^{2}{\left(x \right)}} \cos{\left(x \right)} + 1\right) e^{- a \cos^{2}{\left(x \right)}}\, dx 
=
= 
  pi                                                                              
  --                                                                              
  2                                                                               
   /                                                                              
  |                                                                               
  |  /                   2                                    2   \        2      
  |  |              a*cos (x)                            a*cos (x)|  -a*cos (x)   
  |  \1 + b*cos(x)*e          + b*cos(x)*erfc(b*cos(x))*e         /*e           dx
  |                                                                               
 /                                                                                
-pi                                                                               
----                                                                              
 2
∫−π2π2(beacos⁡2(x)cos⁡(x)erfc⁡(bcos⁡(x))+beacos⁡2(x)cos⁡(x)+1)e−acos⁡2(x) dx\int\limits_{- \frac{\pi}{2}}^{\frac{\pi}{2}} \left(b e^{a \cos^{2}{\left(x \right)}} \cos{\left(x \right)} \operatorname{erfc}{\left(b \cos{\left(x \right)} \right)} + b e^{a \cos^{2}{\left(x \right)}} \cos{\left(x \right)} + 1\right) e^{- a \cos^{2}{\left(x \right)}}\, dx 
Integral((1 + b*cos(x)*exp(a*cos(x)^2) + b*cos(x)*erfc(b*cos(x))*exp(a*cos(x)^2))*exp(-a*cos(x)^2), (x, -pi/2, pi/2))

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

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