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