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Resolución de integrales/ e^(-x)/ e^(-x)sin(u*x)

Integral de e^(-x)sin(u*x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  2                
  /                
 |                 
 |   -x            
 |  E  *sin(u*x) dx
 |                 
/                  
-oo
∫−∞2e−xsin⁡(ux) dx\int\limits_{-\infty}^{2} e^{- x} \sin{\left(u x \right)}\, dx 
Integral(E^(-x)*sin(u*x), (x, -oo, 2))
Respuesta (Indefinida) [src] 
                         //   /         -x              -x      -x        \            \
                         ||   |cosh(x)*e     x*cosh(x)*e     x*e  *sinh(x)|            |
                         ||-I*|----------- + ------------- + -------------|  for u = -I|
                         ||   \     2              2               2      /            |
  /                      ||                                                            |
 |                       ||  /         -x              -x      -x        \             |
 |  -x                   ||  |cosh(x)*e     x*cosh(x)*e     x*e  *sinh(x)|             |
 | E  *sin(u*x) dx = C + |
∫e−xsin⁡(ux) dx=C+{−i(xe−xsinh⁡(x)2+xe−xcosh⁡(x)2+e−xcosh⁡(x)2)for u=−ii(xe−xsinh⁡(x)2+xe−xcosh⁡(x)2+e−xcosh⁡(x)2)for u=i−ucos⁡(ux)u2ex+ex−sin⁡(ux)u2ex+exotherwise\int e^{- x} \sin{\left(u x \right)}\, dx = C + \begin{cases} - i \left(\frac{x e^{- x} \sinh{\left(x \right)}}{2} + \frac{x e^{- x} \cosh{\left(x \right)}}{2} + \frac{e^{- x} \cosh{\left(x \right)}}{2}\right) & \text{for}\: u = - i \\i \left(\frac{x e^{- x} \sinh{\left(x \right)}}{2} + \frac{x e^{- x} \cosh{\left(x \right)}}{2} + \frac{e^{- x} \cosh{\left(x \right)}}{2}\right) & \text{for}\: u = i \\- \frac{u \cos{\left(u x \right)}}{u^{2} e^{x} + e^{x}} - \frac{\sin{\left(u x \right)}}{u^{2} e^{x} + e^{x}} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
       /  1   \                                     sin(2*u)    u*cos(2*u)
oo*sign|------|*sign(-sin(zoo*u) + u*cos(zoo*u)) - ---------- - ----------
       |     2|                                     2  2    2    2  2    2
       \1 + u /                                    u *e  + e    u *e  + e
−ucos⁡(2u)u2e2+e2+∞sign⁡(1u2+1)sign⁡(ucos⁡(∞~u)−sin⁡(∞~u))−sin⁡(2u)u2e2+e2- \frac{u \cos{\left(2 u \right)}}{u^{2} e^{2} + e^{2}} + \infty \operatorname{sign}{\left(\frac{1}{u^{2} + 1} \right)} \operatorname{sign}{\left(u \cos{\left(\tilde{\infty} u \right)} - \sin{\left(\tilde{\infty} u \right)} \right)} - \frac{\sin{\left(2 u \right)}}{u^{2} e^{2} + e^{2}} 
=
= 
       /  1   \                                     sin(2*u)    u*cos(2*u)
oo*sign|------|*sign(-sin(zoo*u) + u*cos(zoo*u)) - ---------- - ----------
       |     2|                                     2  2    2    2  2    2
       \1 + u /                                    u *e  + e    u *e  + e
−ucos⁡(2u)u2e2+e2+∞sign⁡(1u2+1)sign⁡(ucos⁡(∞~u)−sin⁡(∞~u))−sin⁡(2u)u2e2+e2- \frac{u \cos{\left(2 u \right)}}{u^{2} e^{2} + e^{2}} + \infty \operatorname{sign}{\left(\frac{1}{u^{2} + 1} \right)} \operatorname{sign}{\left(u \cos{\left(\tilde{\infty} u \right)} - \sin{\left(\tilde{\infty} u \right)} \right)} - \frac{\sin{\left(2 u \right)}}{u^{2} e^{2} + e^{2}} 
oo*sign(1/(1 + u^2))*sign(-sin(±oo*u) + u*cos(±oo*u)) - sin(2*u)/(u^2*exp(2) + exp(2)) - u*cos(2*u)/(u^2*exp(2) + exp(2))

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

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