Sr Examen
Integral de t*cos(t)*e^(-st) dt
Solución
Ha introducido
[src]
oo / | | -s*t | t*cos(t)*E dt | / 0
$$\int\limits_{0}^{\infty} e^{- s t} t \cos{\left(t \right)}\, dt$$
Integral((t*cos(t))*E^((-s)*t), (t, 0, oo))
Respuesta (Indefinida)
[src]
// I*t I*t 2 I*t I*t 2 I*t \
|| cos(t)*e t*e *sin(t) t *cos(t)*e I*t*cos(t)*e I*t *e *sin(t) |
|| ----------- + ------------- + -------------- - --------------- - ---------------- for s = -I|
|| 4 4 4 4 4 |
/ || |
| || -I*t -I*t 2 -I*t -I*t 2 -I*t |
| -s*t || cos(t)*e t*e *sin(t) t *cos(t)*e I*t*cos(t)*e I*t *e *sin(t) |
| t*cos(t)*E dt = C + |< ------------ + -------------- + --------------- + ---------------- + ----------------- for s = I |
| || 4 4 4 4 4 |
/ || |
|| 2 2 3 |
|| cos(t) t*sin(t) s *cos(t) 2*s*sin(t) t*s *sin(t) s*t*cos(t) t*s *cos(t) |
||-------------------------- + -------------------------- - -------------------------- + -------------------------- + -------------------------- - -------------------------- - -------------------------- otherwise |
|| 4 s*t 2 s*t s*t 4 s*t 2 s*t s*t 4 s*t 2 s*t s*t 4 s*t 2 s*t s*t 4 s*t 2 s*t s*t 4 s*t 2 s*t s*t 4 s*t 2 s*t s*t |
\\s *e + 2*s *e + e s *e + 2*s *e + e s *e + 2*s *e + e s *e + 2*s *e + e s *e + 2*s *e + e s *e + 2*s *e + e s *e + 2*s *e + e /
$$\int e^{- s t} t \cos{\left(t \right)}\, dt = C + \begin{cases} - \frac{i t^{2} e^{i t} \sin{\left(t \right)}}{4} + \frac{t^{2} e^{i t} \cos{\left(t \right)}}{4} + \frac{t e^{i t} \sin{\left(t \right)}}{4} - \frac{i t e^{i t} \cos{\left(t \right)}}{4} + \frac{e^{i t} \cos{\left(t \right)}}{4} & \text{for}\: s = - i \\\frac{i t^{2} e^{- i t} \sin{\left(t \right)}}{4} + \frac{t^{2} e^{- i t} \cos{\left(t \right)}}{4} + \frac{t e^{- i t} \sin{\left(t \right)}}{4} + \frac{i t e^{- i t} \cos{\left(t \right)}}{4} + \frac{e^{- i t} \cos{\left(t \right)}}{4} & \text{for}\: s = i \\- \frac{s^{3} t \cos{\left(t \right)}}{s^{4} e^{s t} + 2 s^{2} e^{s t} + e^{s t}} + \frac{s^{2} t \sin{\left(t \right)}}{s^{4} e^{s t} + 2 s^{2} e^{s t} + e^{s t}} - \frac{s^{2} \cos{\left(t \right)}}{s^{4} e^{s t} + 2 s^{2} e^{s t} + e^{s t}} - \frac{s t \cos{\left(t \right)}}{s^{4} e^{s t} + 2 s^{2} e^{s t} + e^{s t}} + \frac{2 s \sin{\left(t \right)}}{s^{4} e^{s t} + 2 s^{2} e^{s t} + e^{s t}} + \frac{t \sin{\left(t \right)}}{s^{4} e^{s t} + 2 s^{2} e^{s t} + e^{s t}} + \frac{\cos{\left(t \right)}}{s^{4} e^{s t} + 2 s^{2} e^{s t} + e^{s t}} & \text{otherwise} \end{cases}$$
Respuesta
[src]
/ / 2\ | 2*\1 - s / |-------------------- for 2*|arg(s)| < pi |/ 2\ / 2\ |\1 + s /*\-2 - 2*s / | | oo < / | | | | -s*t | | t*cos(t)*e dt otherwise | | |/ |0 \
$$\begin{cases} \frac{2 \left(1 - s^{2}\right)}{\left(- 2 s^{2} - 2\right) \left(s^{2} + 1\right)} & \text{for}\: 2 \left|{\arg{\left(s \right)}}\right| < \pi \\\int\limits_{0}^{\infty} t e^{- s t} \cos{\left(t \right)}\, dt & \text{otherwise} \end{cases}$$
=
=
/ / 2\ | 2*\1 - s / |-------------------- for 2*|arg(s)| < pi |/ 2\ / 2\ |\1 + s /*\-2 - 2*s / | | oo < / | | | | -s*t | | t*cos(t)*e dt otherwise | | |/ |0 \
$$\begin{cases} \frac{2 \left(1 - s^{2}\right)}{\left(- 2 s^{2} - 2\right) \left(s^{2} + 1\right)} & \text{for}\: 2 \left|{\arg{\left(s \right)}}\right| < \pi \\\int\limits_{0}^{\infty} t e^{- s t} \cos{\left(t \right)}\, dt & \text{otherwise} \end{cases}$$
Piecewise((2*(1 - s^2)/((1 + s^2)*(-2 - 2*s^2)), 2*Abs(arg(s)) < pi), (Integral(t*cos(t)*exp(-s*t), (t, 0, oo)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.