Sr Examen
Integral de exp(-n*(t-x))*sin(k*(t-x)) dx
Solución
Respuesta
[src]
// /1 \
|| |-- for k != 0 |
/ // /cos(k*t) \ \ || | 2 |
| || |-------- for k != 0 | | || |k |
| || | 2 | | || < for n = 0|
| || < k for n = 0| | || | 2 |
| || | | | || |t |
| || | 0 otherwise | | || |-- otherwise |
| || \ | | || \2 |
| || | | || |
| ||/ 0 for And(k = 0, n = 0) | //0 for n = 0\ | ||/ t for And(k = 0, n = 0) |
| ||| | || | | -n*t ||| | -n*t
- |k*|<| -sinh(n*t) | + |<1 |*sin(k*t)|*e + k*|<| n*t |*e
| ||| ----------- for Or(k = -I*n, k = I*n) | ||- otherwise| | |||t*e |
| ||| 2*n | \\n / | |||------ for Or(k = -I*n, k = I*n) |
| ||< | | ||| 2 |
| |||n*cos(k*t) k*sin(k*t) | | ||< |
| |||---------- - ---------- otherwise | | ||| n*t |
| ||| 2 2 2 2 | | ||| n*e |
| ||| k + n k + n | | |||------- otherwise |
| ||\ | | ||| 2 2 |
| ||--------------------------------------------------- otherwise| | |||k + n |
\ \\ n / / ||\ |
||----------------------------------- otherwise|
\\ n /
$$k \left(\begin{cases} \begin{cases} \frac{1}{k^{2}} & \text{for}\: k \neq 0 \\\frac{t^{2}}{2} & \text{otherwise} \end{cases} & \text{for}\: n = 0 \\\frac{\begin{cases} t & \text{for}\: k = 0 \wedge n = 0 \\\frac{t e^{n t}}{2} & \text{for}\: k = - i n \vee k = i n \\\frac{n e^{n t}}{k^{2} + n^{2}} & \text{otherwise} \end{cases}}{n} & \text{otherwise} \end{cases}\right) e^{- n t} - \left(k \left(\begin{cases} \begin{cases} \frac{\cos{\left(k t \right)}}{k^{2}} & \text{for}\: k \neq 0 \\0 & \text{otherwise} \end{cases} & \text{for}\: n = 0 \\\frac{\begin{cases} 0 & \text{for}\: k = 0 \wedge n = 0 \\- \frac{\sinh{\left(n t \right)}}{2 n} & \text{for}\: k = - i n \vee k = i n \\- \frac{k \sin{\left(k t \right)}}{k^{2} + n^{2}} + \frac{n \cos{\left(k t \right)}}{k^{2} + n^{2}} & \text{otherwise} \end{cases}}{n} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: n = 0 \\\frac{1}{n} & \text{otherwise} \end{cases}\right) \sin{\left(k t \right)}\right) e^{- n t}$$
=
=
// /1 \
|| |-- for k != 0 |
/ // /cos(k*t) \ \ || | 2 |
| || |-------- for k != 0 | | || |k |
| || | 2 | | || < for n = 0|
| || < k for n = 0| | || | 2 |
| || | | | || |t |
| || | 0 otherwise | | || |-- otherwise |
| || \ | | || \2 |
| || | | || |
| ||/ 0 for And(k = 0, n = 0) | //0 for n = 0\ | ||/ t for And(k = 0, n = 0) |
| ||| | || | | -n*t ||| | -n*t
- |k*|<| -sinh(n*t) | + |<1 |*sin(k*t)|*e + k*|<| n*t |*e
| ||| ----------- for Or(k = -I*n, k = I*n) | ||- otherwise| | |||t*e |
| ||| 2*n | \\n / | |||------ for Or(k = -I*n, k = I*n) |
| ||< | | ||| 2 |
| |||n*cos(k*t) k*sin(k*t) | | ||< |
| |||---------- - ---------- otherwise | | ||| n*t |
| ||| 2 2 2 2 | | ||| n*e |
| ||| k + n k + n | | |||------- otherwise |
| ||\ | | ||| 2 2 |
| ||--------------------------------------------------- otherwise| | |||k + n |
\ \\ n / / ||\ |
||----------------------------------- otherwise|
\\ n /
$$k \left(\begin{cases} \begin{cases} \frac{1}{k^{2}} & \text{for}\: k \neq 0 \\\frac{t^{2}}{2} & \text{otherwise} \end{cases} & \text{for}\: n = 0 \\\frac{\begin{cases} t & \text{for}\: k = 0 \wedge n = 0 \\\frac{t e^{n t}}{2} & \text{for}\: k = - i n \vee k = i n \\\frac{n e^{n t}}{k^{2} + n^{2}} & \text{otherwise} \end{cases}}{n} & \text{otherwise} \end{cases}\right) e^{- n t} - \left(k \left(\begin{cases} \begin{cases} \frac{\cos{\left(k t \right)}}{k^{2}} & \text{for}\: k \neq 0 \\0 & \text{otherwise} \end{cases} & \text{for}\: n = 0 \\\frac{\begin{cases} 0 & \text{for}\: k = 0 \wedge n = 0 \\- \frac{\sinh{\left(n t \right)}}{2 n} & \text{for}\: k = - i n \vee k = i n \\- \frac{k \sin{\left(k t \right)}}{k^{2} + n^{2}} + \frac{n \cos{\left(k t \right)}}{k^{2} + n^{2}} & \text{otherwise} \end{cases}}{n} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: n = 0 \\\frac{1}{n} & \text{otherwise} \end{cases}\right) \sin{\left(k t \right)}\right) e^{- n t}$$
-(k*Piecewise((Piecewise((cos(k*t)/k^2, Ne(k, 0)), (0, True)), n = 0), (Piecewise((0, (k = 0)∧(n = 0)), (-sinh(n*t)/(2*n), (k = i*n)∨(k = -i*n)), (n*cos(k*t)/(k^2 + n^2) - k*sin(k*t)/(k^2 + n^2), True))/n, True)) + Piecewise((0, n = 0), (1/n, True))*sin(k*t))*exp(-n*t) + k*Piecewise((Piecewise((k^(-2), Ne(k, 0)), (t^2/2, True)), n = 0), (Piecewise((t, (k = 0)∧(n = 0)), (t*exp(n*t)/2, (k = i*n)∨(k = -i*n)), (n*exp(n*t)/(k^2 + n^2), True))/n, True))*exp(-n*t)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.