Sr Examen
Integral de cos(w*t)*exp(-i*w*t) dt
Solución
Ha introducido
[src]
t2 / | | -I*w*t | cos(w*t)*e dt | / 0
$$\int\limits_{0}^{t_{2}} e^{t - i w} \cos{\left(t w \right)}\, dt$$
Integral(cos(w*t)*exp(((-i)*w)*t), (t, 0, t2))
Respuesta (Indefinida)
[src]
/// /t*w\ 2/t*w\ \
||| 2*tan|---| t*w*tan |---| |
||| \ 2 / t*w \ 2 / |
|||----------------- - ----------------- + ----------------- for w != 0 |
||< 2 2 2/t*w\ 2 2 2/t*w\ 2 2 2/t*w\ for w = 0|
|||w + w *tan |---| w + w *tan |---| w + w *tan |---| |
/ ||| \ 2 / \ 2 / \ 2 / | // t for w = 0\
| ||| | || |
| -I*w*t ||\ 0 otherwise | || -I*t*w |
| cos(w*t)*e dt = C + w*|< | + |
$$\int e^{t - i w} \cos{\left(t w \right)}\, dt = C + w \left(\begin{cases} \begin{cases} \frac{t w \tan^{2}{\left(\frac{t w}{2} \right)}}{w^{2} \tan^{2}{\left(\frac{t w}{2} \right)} + w^{2}} - \frac{t w}{w^{2} \tan^{2}{\left(\frac{t w}{2} \right)} + w^{2}} + \frac{2 \tan{\left(\frac{t w}{2} \right)}}{w^{2} \tan^{2}{\left(\frac{t w}{2} \right)} + w^{2}} & \text{for}\: w \neq 0 \\0 & \text{otherwise} \end{cases} & \text{for}\: w = 0 \\\frac{t}{2 w} + \begin{cases} - \frac{i e^{- 2 i t w}}{4 w^{2}} & \text{for}\: 4 w^{2} \neq 0 \\- \frac{t}{2 w} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} t & \text{for}\: w = 0 \\\frac{i e^{- i t w}}{w} & \text{otherwise} \end{cases}\right) \cos{\left(t w \right)}$$
Respuesta
[src]
// -2*I*t2*w \
|| I I*e |
||- --- + ------------ for And(w > -oo, w < oo, w != 0)|
t2 || 4*w 4*w |
-- + |< |
2 || t2 |
|| -- otherwise |
|| 2 |
\\ /
$$\frac{t_{2}}{2} + \begin{cases} - \frac{i}{4 w} + \frac{i e^{- 2 i t_{2} w}}{4 w} & \text{for}\: w > -\infty \wedge w < \infty \wedge w \neq 0 \\\frac{t_{2}}{2} & \text{otherwise} \end{cases}$$
=
=
// -2*I*t2*w \
|| I I*e |
||- --- + ------------ for And(w > -oo, w < oo, w != 0)|
t2 || 4*w 4*w |
-- + |< |
2 || t2 |
|| -- otherwise |
|| 2 |
\\ /
$$\frac{t_{2}}{2} + \begin{cases} - \frac{i}{4 w} + \frac{i e^{- 2 i t_{2} w}}{4 w} & \text{for}\: w > -\infty \wedge w < \infty \wedge w \neq 0 \\\frac{t_{2}}{2} & \text{otherwise} \end{cases}$$
t2/2 + Piecewise((-i/(4*w) + i*exp(-2*i*t2*w)/(4*w), (w > -oo)∧(w < oo)∧(Ne(w, 0))), (t2/2, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.