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

Integral de exp(-a*x^2)*cos(b*x^2) 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⁡(bx2) dx\int\limits_{0}^{\infty} e^{- a x^{2}} \cos{\left(b x^{2} \right)}\, dx 
Integral(exp((-a)*x^2)*cos(b*x^2), (x, 0, oo))
Respuesta [src] 
/         /    /      ____\\                                                                                                                    
|         |    |     /  2 ||                                                                                                                    
|         |    |    /  b  ||                                                                                                                    
|         |atan|   /   -- ||                                                                                                                    
|         |    |  /     2 ||                                                                                                                    
|  ___    |    \\/     a  /|                                                                                                                    
|\/ 2 *cos|----------------|*Gamma(1/4)*Gamma(3/4)        /                                         /                                  2      \\
|         \       2        /                              |                                         |                                 a       ||
|-------------------------------------------------  for Or|And(2*|arg(b)| = 0, 2*|arg(a)| < pi), And|2*|arg(a)| = pi, 2*|arg(b)| = 0, -- != -1||
|                               ________                  |                                         |                                  2      ||
|                              /      2                   \                                         \                                 b       //
|              ____   ___     /      b                                                                                                          
<          4*\/ pi *\/ a *   /   1 + --                                                                                                         
|                         4 /         2                                                                                                         
|                         \/         a                                                                                                          
|                                                                                                                                               
|              oo                                                                                                                               
|               /                                                                                                                               
|              |                                                                                                                                
|              |                 2                                                                                                              
|              |     /   2\  -a*x                                                                                                               
|              |  cos\b*x /*e      dx                                                        otherwise                                          
|              |                                                                                                                                
|             /                                                                                                                                 
\             0
{2cos⁡(atan⁡(b2a2)2)Γ(14)Γ(34)4πa1+b2a24for (2∣arg⁡(b)∣=0∧2∣arg⁡(a)∣<π)∨(2∣arg⁡(a)∣=π∧2∣arg⁡(b)∣=0∧a2b2≠−1)∫0∞e−ax2cos⁡(bx2) dxotherwise\begin{cases} \frac{\sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{\frac{b^{2}}{a^{2}}} \right)}}{2} \right)} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right)}{4 \sqrt{\pi} \sqrt{a} \sqrt[4]{1 + \frac{b^{2}}{a^{2}}}} & \text{for}\: \left(2 \left|{\arg{\left(b \right)}}\right| = 0 \wedge 2 \left|{\arg{\left(a \right)}}\right| < \pi\right) \vee \left(2 \left|{\arg{\left(a \right)}}\right| = \pi \wedge 2 \left|{\arg{\left(b \right)}}\right| = 0 \wedge \frac{a^{2}}{b^{2}} \neq -1\right) \\\int\limits_{0}^{\infty} e^{- a x^{2}} \cos{\left(b x^{2} \right)}\, dx & \text{otherwise} \end{cases} 
=
= 
/         /    /      ____\\                                                                                                                    
|         |    |     /  2 ||                                                                                                                    
|         |    |    /  b  ||                                                                                                                    
|         |atan|   /   -- ||                                                                                                                    
|         |    |  /     2 ||                                                                                                                    
|  ___    |    \\/     a  /|                                                                                                                    
|\/ 2 *cos|----------------|*Gamma(1/4)*Gamma(3/4)        /                                         /                                  2      \\
|         \       2        /                              |                                         |                                 a       ||
|-------------------------------------------------  for Or|And(2*|arg(b)| = 0, 2*|arg(a)| < pi), And|2*|arg(a)| = pi, 2*|arg(b)| = 0, -- != -1||
|                               ________                  |                                         |                                  2      ||
|                              /      2                   \                                         \                                 b       //
|              ____   ___     /      b                                                                                                          
<          4*\/ pi *\/ a *   /   1 + --                                                                                                         
|                         4 /         2                                                                                                         
|                         \/         a                                                                                                          
|                                                                                                                                               
|              oo                                                                                                                               
|               /                                                                                                                               
|              |                                                                                                                                
|              |                 2                                                                                                              
|              |     /   2\  -a*x                                                                                                               
|              |  cos\b*x /*e      dx                                                        otherwise                                          
|              |                                                                                                                                
|             /                                                                                                                                 
\             0
{2cos⁡(atan⁡(b2a2)2)Γ(14)Γ(34)4πa1+b2a24for (2∣arg⁡(b)∣=0∧2∣arg⁡(a)∣<π)∨(2∣arg⁡(a)∣=π∧2∣arg⁡(b)∣=0∧a2b2≠−1)∫0∞e−ax2cos⁡(bx2) dxotherwise\begin{cases} \frac{\sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{\frac{b^{2}}{a^{2}}} \right)}}{2} \right)} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right)}{4 \sqrt{\pi} \sqrt{a} \sqrt[4]{1 + \frac{b^{2}}{a^{2}}}} & \text{for}\: \left(2 \left|{\arg{\left(b \right)}}\right| = 0 \wedge 2 \left|{\arg{\left(a \right)}}\right| < \pi\right) \vee \left(2 \left|{\arg{\left(a \right)}}\right| = \pi \wedge 2 \left|{\arg{\left(b \right)}}\right| = 0 \wedge \frac{a^{2}}{b^{2}} \neq -1\right) \\\int\limits_{0}^{\infty} e^{- a x^{2}} \cos{\left(b x^{2} \right)}\, dx & \text{otherwise} \end{cases} 
Piecewise((sqrt(2)*cos(atan(sqrt(b^2/a^2))/2)*gamma(1/4)*gamma(3/4)/(4*sqrt(pi)*sqrt(a)*(1 + b^2/a^2)^(1/4)), ((2*Abs(arg(b)) = 0))∧(2*Abs(arg(a)) < pi))∨((Eq(2*Abs(arg(a), pi))∧(2*Abs(arg(b)) = 0)∧(Ne(a^2/b^2, -1)))), (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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