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