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