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