Integral de e^(-x)sin(u*x) dx
Solución
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 + |
$$\int e^{- x} \sin{\left(u x \right)}\, dx = C + \begin{cases} - i \left(\frac{x e^{- x} \sinh{\left(x \right)}}{2} + \frac{x e^{- x} \cosh{\left(x \right)}}{2} + \frac{e^{- x} \cosh{\left(x \right)}}{2}\right) & \text{for}\: u = - i \\i \left(\frac{x e^{- x} \sinh{\left(x \right)}}{2} + \frac{x e^{- x} \cosh{\left(x \right)}}{2} + \frac{e^{- x} \cosh{\left(x \right)}}{2}\right) & \text{for}\: u = i \\- \frac{u \cos{\left(u x \right)}}{u^{2} e^{x} + e^{x}} - \frac{\sin{\left(u x \right)}}{u^{2} e^{x} + e^{x}} & \text{otherwise} \end{cases}$$
/ 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
$$- \frac{u \cos{\left(2 u \right)}}{u^{2} e^{2} + e^{2}} + \infty \operatorname{sign}{\left(\frac{1}{u^{2} + 1} \right)} \operatorname{sign}{\left(u \cos{\left(\tilde{\infty} u \right)} - \sin{\left(\tilde{\infty} u \right)} \right)} - \frac{\sin{\left(2 u \right)}}{u^{2} e^{2} + e^{2}}$$
=
/ 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
$$- \frac{u \cos{\left(2 u \right)}}{u^{2} e^{2} + e^{2}} + \infty \operatorname{sign}{\left(\frac{1}{u^{2} + 1} \right)} \operatorname{sign}{\left(u \cos{\left(\tilde{\infty} u \right)} - \sin{\left(\tilde{\infty} u \right)} \right)} - \frac{\sin{\left(2 u \right)}}{u^{2} e^{2} + e^{2}}$$
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.