Sr Examen
Integral de e^(-x)*sin(w*x) dx
Solución
Ha introducido
[src]
oo / | | -x | E *sin(w*x) dx | / 0
$$\int\limits_{0}^{\infty} e^{- x} \sin{\left(w x \right)}\, dx$$
Integral(E^(-x)*sin(w*x), (x, 0, oo))
Respuesta (Indefinida)
[src]
// / -x -x -x \ \
|| |cosh(x)*e x*cosh(x)*e x*e *sinh(x)| |
||-I*|----------- + ------------- + -------------| for w = -I|
|| \ 2 2 2 / |
/ || |
| || / -x -x -x \ |
| -x || |cosh(x)*e x*cosh(x)*e x*e *sinh(x)| |
| E *sin(w*x) dx = C + |
$$\int e^{- x} \sin{\left(w x \right)}\, dx = C + \begin{cases} - i \left(\frac{x e^{- x} \sinh{\left(x \right)}}{2} + \frac{x e^{- x} \cosh{\left(x \right)}}{2} + \frac{e^{- x} \cosh{\left(x \right)}}{2}\right) & \text{for}\: w = - i \\i \left(\frac{x e^{- x} \sinh{\left(x \right)}}{2} + \frac{x e^{- x} \cosh{\left(x \right)}}{2} + \frac{e^{- x} \cosh{\left(x \right)}}{2}\right) & \text{for}\: w = i \\- \frac{w \cos{\left(w x \right)}}{w^{2} e^{x} + e^{x}} - \frac{\sin{\left(w x \right)}}{w^{2} e^{x} + e^{x}} & \text{otherwise} \end{cases}$$
Respuesta
[src]
/ w | ------ for 2*|arg(w)| = 0 | 2 | 1 + w | | oo < / | | | | -x | | e *sin(w*x) dx otherwise | | |/ \0
$$\begin{cases} \frac{w}{w^{2} + 1} & \text{for}\: 2 \left|{\arg{\left(w \right)}}\right| = 0 \\\int\limits_{0}^{\infty} e^{- x} \sin{\left(w x \right)}\, dx & \text{otherwise} \end{cases}$$
=
=
/ w | ------ for 2*|arg(w)| = 0 | 2 | 1 + w | | oo < / | | | | -x | | e *sin(w*x) dx otherwise | | |/ \0
$$\begin{cases} \frac{w}{w^{2} + 1} & \text{for}\: 2 \left|{\arg{\left(w \right)}}\right| = 0 \\\int\limits_{0}^{\infty} e^{- x} \sin{\left(w x \right)}\, dx & \text{otherwise} \end{cases}$$
Piecewise((w/(1 + w^2), 2*Abs(arg(w)) = 0), (Integral(exp(-x)*sin(w*x), (x, 0, oo)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.