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Resolución de integrales/ (cos(6*x)-(1/(Pi))*(sin(6*x)))*e^(5*x-a*x)

Integral de (cos(6*x)-(1/(Pi))*(sin(6*x)))*e^(5*x-a*x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  0                                    
  /                                    
 |                                     
 |  /           sin(6*x)\  5*x - a*x   
 |  |cos(6*x) - --------|*E          dx
 |  \              pi   /              
 |                                     
/                                      
-oo
∫−∞0e−ax+5x(−sin⁡(6x)π+cos⁡(6x)) dx\int\limits_{-\infty}^{0} e^{- a x + 5 x} \left(- \frac{\sin{\left(6 x \right)}}{\pi} + \cos{\left(6 x \right)}\right)\, dx 
Integral((cos(6*x) - sin(6*x)/pi)*E^(5*x - a*x), (x, -oo, 0))
Respuesta [src] 
/                               /5   a\                                
|                             4*|- - -|                                
|         1                     \4   4/                                
|------------------- + -----------------------  for 2*|arg(5 - a)| < pi
|     /           2\   /       36   \        2                         
|     |    (5 - a) |   |1 + --------|*(5 - a)                          
|6*pi*|1 + --------|   |           2|                                  
|     \       36   /   \    (5 - a) /                                  
|                                                                      
<    0                                                                 
|    /                                                                 
|   |                                                                  
|   |  /  sin(6*x)           \  5*x - a*x                              
|   |  |- -------- + cos(6*x)|*e          dx           otherwise       
|   |  \     pi              /                                         
|   |                                                                  
|  /                                                                   
|  -oo                                                                 
\
{16π((5−a)236+1)+4(54−a4)(1+36(5−a)2)(5−a)2for 2∣arg⁡(5−a)∣<π∫−∞0(−sin⁡(6x)π+cos⁡(6x))e−ax+5x dxotherwise\begin{cases} \frac{1}{6 \pi \left(\frac{\left(5 - a\right)^{2}}{36} + 1\right)} + \frac{4 \left(\frac{5}{4} - \frac{a}{4}\right)}{\left(1 + \frac{36}{\left(5 - a\right)^{2}}\right) \left(5 - a\right)^{2}} & \text{for}\: 2 \left|{\arg{\left(5 - a \right)}}\right| < \pi \\\int\limits_{-\infty}^{0} \left(- \frac{\sin{\left(6 x \right)}}{\pi} + \cos{\left(6 x \right)}\right) e^{- a x + 5 x}\, dx & \text{otherwise} \end{cases} 
=
= 
/                               /5   a\                                
|                             4*|- - -|                                
|         1                     \4   4/                                
|------------------- + -----------------------  for 2*|arg(5 - a)| < pi
|     /           2\   /       36   \        2                         
|     |    (5 - a) |   |1 + --------|*(5 - a)                          
|6*pi*|1 + --------|   |           2|                                  
|     \       36   /   \    (5 - a) /                                  
|                                                                      
<    0                                                                 
|    /                                                                 
|   |                                                                  
|   |  /  sin(6*x)           \  5*x - a*x                              
|   |  |- -------- + cos(6*x)|*e          dx           otherwise       
|   |  \     pi              /                                         
|   |                                                                  
|  /                                                                   
|  -oo                                                                 
\
{16π((5−a)236+1)+4(54−a4)(1+36(5−a)2)(5−a)2for 2∣arg⁡(5−a)∣<π∫−∞0(−sin⁡(6x)π+cos⁡(6x))e−ax+5x dxotherwise\begin{cases} \frac{1}{6 \pi \left(\frac{\left(5 - a\right)^{2}}{36} + 1\right)} + \frac{4 \left(\frac{5}{4} - \frac{a}{4}\right)}{\left(1 + \frac{36}{\left(5 - a\right)^{2}}\right) \left(5 - a\right)^{2}} & \text{for}\: 2 \left|{\arg{\left(5 - a \right)}}\right| < \pi \\\int\limits_{-\infty}^{0} \left(- \frac{\sin{\left(6 x \right)}}{\pi} + \cos{\left(6 x \right)}\right) e^{- a x + 5 x}\, dx & \text{otherwise} \end{cases} 
Piecewise((1/(6*pi*(1 + (5 - a)^2/36)) + 4*(5/4 - a/4)/((1 + 36/(5 - a)^2)*(5 - a)^2), 2*Abs(arg(5 - a)) < pi), (Integral((-sin(6*x)/pi + cos(6*x))*exp(5*x - a*x), (x, -oo, 0)), True))

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

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