Sr Examen
Integral de e^(-(x-y)^2/(4t)-|y|) dy
Solución
Ha introducido
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oo / | | 2 | -(x - y) | ---------- - |y| | 4*t | E dy | / -oo
$$\int\limits_{-\infty}^{\infty} e^{- \left|{y}\right| + \frac{\left(-1\right) \left(x - y\right)^{2}}{4 t}}\, dy$$
Integral(E^((-(x - y)^2)/((4*t)) - |y|), (y, -oo, oo))
Solución detallada
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Vuelva a escribir el integrando:
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La integral del producto de una función por una constante es la constante por la integral de esta función:
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No puedo encontrar los pasos en la búsqueda de esta integral.
Pero la integral
Por lo tanto, el resultado es:
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Añadimos la constante de integración:
Respuesta:
Respuesta (Indefinida)
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/ / / \ | | | | | 2 | | 2 | 2 | -(x - y) | | -y x*y | -x | ---------- - |y| | | ---- --- | ---- | 4*t | | -|y| 4*t 2*t | 4*t | E dy = C + | | e *e *e dy|*e | | | | / \/ /
$$\int e^{- \left|{y}\right| + \frac{\left(-1\right) \left(x - y\right)^{2}}{4 t}}\, dy = C + e^{- \frac{x^{2}}{4 t}} \int e^{- \frac{y^{2}}{4 t}} e^{\frac{x y}{2 t}} e^{- \left|{y}\right|}\, dy$$
Respuesta
[src]
/ / 2 2\ | | / x\ / x\ | | | t*|2 + -| t*|2 + -| | | | \ t/ / ___ / x\\ \ t/ | |/ 2 2\ 2 | ---------- |\/ t *|2 + -|| ----------| 2 || / x \ / x \ | -x | ___ / x\ 4 ___ / x\ | \ t/| 4 | -x || t*|1 - ---| t*|1 - ---| | ---- |pi*\/ t *|2 + -|*e pi*\/ t *|2 + -|*erf|-------------|*e | ---- || ___ / x \ \ 2*t/ ___ / x \ / ___ / x \\ \ 2*t/ | 4*t | \ t/ \ t/ \ 2 / | 4*t ||pi*\/ t *|1 - ---|*e - pi*\/ t *|1 - ---|*erf|\/ t *|1 - ---||*e |*e 2*|---------------------------- - -----------------------------------------------|*e |\ \ 2*t/ \ 2*t/ \ \ 2*t// / \ 2 2 / / / / pi | / x \| \ / | / x \| pi\ / pi | / x \| \ pi\ / / pi | / x\| \ / | / x\| pi\ / pi | / x\| \ pi\\ |------------------------------------------------------------------------------------------------ + ---------------------------------------------------------------------------------------- for And|Or|And||arg(t)| <= --, 2*|arg|1 - ---|| < pi|, And|2*|arg|1 - ---|| <= pi, |arg(t)| < --|, And||arg(t)| < --, 2*|arg|1 - ---|| < pi|, |arg(t)| < --|, Or|And||arg(t)| <= --, 2*|arg|2 + -|| < pi|, And|2*|arg|2 + -|| <= pi, |arg(t)| < --|, And||arg(t)| < --, 2*|arg|2 + -|| < pi|, |arg(t)| < --|| | ____ / x \ ____ / x\ \ \ \ 2 | \ 2*t/| / \ | \ 2*t/| 2 / \ 2 | \ 2*t/| / 2 / \ \ 2 | \ t/| / \ | \ t/| 2 / \ 2 | \ t/| / 2 // | \/ pi *|1 - ---| \/ pi *|2 + -| < \ 2*t/ \ t/ | | oo | / | | | | 2 | | (x - y) | | -|y| - -------- | | 4*t | | e dy otherwise | | | / | -oo \
$$\begin{cases} \frac{2 \left(- \frac{\pi \sqrt{t} \left(2 + \frac{x}{t}\right) e^{\frac{t \left(2 + \frac{x}{t}\right)^{2}}{4}} \operatorname{erf}{\left(\frac{\sqrt{t} \left(2 + \frac{x}{t}\right)}{2} \right)}}{2} + \frac{\pi \sqrt{t} \left(2 + \frac{x}{t}\right) e^{\frac{t \left(2 + \frac{x}{t}\right)^{2}}{4}}}{2}\right) e^{- \frac{x^{2}}{4 t}}}{\sqrt{\pi} \left(2 + \frac{x}{t}\right)} + \frac{\left(- \pi \sqrt{t} \left(1 - \frac{x}{2 t}\right) e^{t \left(1 - \frac{x}{2 t}\right)^{2}} \operatorname{erf}{\left(\sqrt{t} \left(1 - \frac{x}{2 t}\right) \right)} + \pi \sqrt{t} \left(1 - \frac{x}{2 t}\right) e^{t \left(1 - \frac{x}{2 t}\right)^{2}}\right) e^{- \frac{x^{2}}{4 t}}}{\sqrt{\pi} \left(1 - \frac{x}{2 t}\right)} & \text{for}\: \left(\left(\left|{\arg{\left(t \right)}}\right| \leq \frac{\pi}{2} \wedge 2 \left|{\arg{\left(1 - \frac{x}{2 t} \right)}}\right| < \pi\right) \vee \left(2 \left|{\arg{\left(1 - \frac{x}{2 t} \right)}}\right| \leq \pi \wedge \left|{\arg{\left(t \right)}}\right| < \frac{\pi}{2}\right) \vee \left(\left|{\arg{\left(t \right)}}\right| < \frac{\pi}{2} \wedge 2 \left|{\arg{\left(1 - \frac{x}{2 t} \right)}}\right| < \pi\right) \vee \left|{\arg{\left(t \right)}}\right| < \frac{\pi}{2}\right) \wedge \left(\left(\left|{\arg{\left(t \right)}}\right| \leq \frac{\pi}{2} \wedge 2 \left|{\arg{\left(2 + \frac{x}{t} \right)}}\right| < \pi\right) \vee \left(2 \left|{\arg{\left(2 + \frac{x}{t} \right)}}\right| \leq \pi \wedge \left|{\arg{\left(t \right)}}\right| < \frac{\pi}{2}\right) \vee \left(\left|{\arg{\left(t \right)}}\right| < \frac{\pi}{2} \wedge 2 \left|{\arg{\left(2 + \frac{x}{t} \right)}}\right| < \pi\right) \vee \left|{\arg{\left(t \right)}}\right| < \frac{\pi}{2}\right) \\\int\limits_{-\infty}^{\infty} e^{- \left|{y}\right| - \frac{\left(x - y\right)^{2}}{4 t}}\, dy & \text{otherwise} \end{cases}$$
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/ / 2 2\ | | / x\ / x\ | | | t*|2 + -| t*|2 + -| | | | \ t/ / ___ / x\\ \ t/ | |/ 2 2\ 2 | ---------- |\/ t *|2 + -|| ----------| 2 || / x \ / x \ | -x | ___ / x\ 4 ___ / x\ | \ t/| 4 | -x || t*|1 - ---| t*|1 - ---| | ---- |pi*\/ t *|2 + -|*e pi*\/ t *|2 + -|*erf|-------------|*e | ---- || ___ / x \ \ 2*t/ ___ / x \ / ___ / x \\ \ 2*t/ | 4*t | \ t/ \ t/ \ 2 / | 4*t ||pi*\/ t *|1 - ---|*e - pi*\/ t *|1 - ---|*erf|\/ t *|1 - ---||*e |*e 2*|---------------------------- - -----------------------------------------------|*e |\ \ 2*t/ \ 2*t/ \ \ 2*t// / \ 2 2 / / / / pi | / x \| \ / | / x \| pi\ / pi | / x \| \ pi\ / / pi | / x\| \ / | / x\| pi\ / pi | / x\| \ pi\\ |------------------------------------------------------------------------------------------------ + ---------------------------------------------------------------------------------------- for And|Or|And||arg(t)| <= --, 2*|arg|1 - ---|| < pi|, And|2*|arg|1 - ---|| <= pi, |arg(t)| < --|, And||arg(t)| < --, 2*|arg|1 - ---|| < pi|, |arg(t)| < --|, Or|And||arg(t)| <= --, 2*|arg|2 + -|| < pi|, And|2*|arg|2 + -|| <= pi, |arg(t)| < --|, And||arg(t)| < --, 2*|arg|2 + -|| < pi|, |arg(t)| < --|| | ____ / x \ ____ / x\ \ \ \ 2 | \ 2*t/| / \ | \ 2*t/| 2 / \ 2 | \ 2*t/| / 2 / \ \ 2 | \ t/| / \ | \ t/| 2 / \ 2 | \ t/| / 2 // | \/ pi *|1 - ---| \/ pi *|2 + -| < \ 2*t/ \ t/ | | oo | / | | | | 2 | | (x - y) | | -|y| - -------- | | 4*t | | e dy otherwise | | | / | -oo \
$$\begin{cases} \frac{2 \left(- \frac{\pi \sqrt{t} \left(2 + \frac{x}{t}\right) e^{\frac{t \left(2 + \frac{x}{t}\right)^{2}}{4}} \operatorname{erf}{\left(\frac{\sqrt{t} \left(2 + \frac{x}{t}\right)}{2} \right)}}{2} + \frac{\pi \sqrt{t} \left(2 + \frac{x}{t}\right) e^{\frac{t \left(2 + \frac{x}{t}\right)^{2}}{4}}}{2}\right) e^{- \frac{x^{2}}{4 t}}}{\sqrt{\pi} \left(2 + \frac{x}{t}\right)} + \frac{\left(- \pi \sqrt{t} \left(1 - \frac{x}{2 t}\right) e^{t \left(1 - \frac{x}{2 t}\right)^{2}} \operatorname{erf}{\left(\sqrt{t} \left(1 - \frac{x}{2 t}\right) \right)} + \pi \sqrt{t} \left(1 - \frac{x}{2 t}\right) e^{t \left(1 - \frac{x}{2 t}\right)^{2}}\right) e^{- \frac{x^{2}}{4 t}}}{\sqrt{\pi} \left(1 - \frac{x}{2 t}\right)} & \text{for}\: \left(\left(\left|{\arg{\left(t \right)}}\right| \leq \frac{\pi}{2} \wedge 2 \left|{\arg{\left(1 - \frac{x}{2 t} \right)}}\right| < \pi\right) \vee \left(2 \left|{\arg{\left(1 - \frac{x}{2 t} \right)}}\right| \leq \pi \wedge \left|{\arg{\left(t \right)}}\right| < \frac{\pi}{2}\right) \vee \left(\left|{\arg{\left(t \right)}}\right| < \frac{\pi}{2} \wedge 2 \left|{\arg{\left(1 - \frac{x}{2 t} \right)}}\right| < \pi\right) \vee \left|{\arg{\left(t \right)}}\right| < \frac{\pi}{2}\right) \wedge \left(\left(\left|{\arg{\left(t \right)}}\right| \leq \frac{\pi}{2} \wedge 2 \left|{\arg{\left(2 + \frac{x}{t} \right)}}\right| < \pi\right) \vee \left(2 \left|{\arg{\left(2 + \frac{x}{t} \right)}}\right| \leq \pi \wedge \left|{\arg{\left(t \right)}}\right| < \frac{\pi}{2}\right) \vee \left(\left|{\arg{\left(t \right)}}\right| < \frac{\pi}{2} \wedge 2 \left|{\arg{\left(2 + \frac{x}{t} \right)}}\right| < \pi\right) \vee \left|{\arg{\left(t \right)}}\right| < \frac{\pi}{2}\right) \\\int\limits_{-\infty}^{\infty} e^{- \left|{y}\right| - \frac{\left(x - y\right)^{2}}{4 t}}\, dy & \text{otherwise} \end{cases}$$
Piecewise(((pi*sqrt(t)*(1 - x/(2*t))*exp(t*(1 - x/(2*t))^2) - pi*sqrt(t)*(1 - x/(2*t))*erf(sqrt(t)*(1 - x/(2*t)))*exp(t*(1 - x/(2*t))^2))*exp(-x^2/(4*t))/(sqrt(pi)*(1 - x/(2*t))) + 2*(pi*sqrt(t)*(2 + x/t)*exp(t*(2 + x/t)^2/4)/2 - pi*sqrt(t)*(2 + x/t)*erf(sqrt(t)*(2 + x/t)/2)*exp(t*(2 + x/t)^2/4)/2)*exp(-x^2/(4*t))/(sqrt(pi)*(2 + x/t)), ((Abs(arg(t)) < pi/2)∨((Abs(arg(t)) <= pi/2)∧(2*Abs(arg(2 + x/t)) < pi))∨((Abs(arg(t)) < pi/2)∧(2*Abs(arg(2 + x/t)) <= pi))∨((Abs(arg(t)) < pi/2)∧(2*Abs(arg(2 + x/t)) < pi)))∧((Abs(arg(t)) < pi/2)∨((Abs(arg(t)) <= pi/2)∧(2*Abs(arg(1 - x/(2*t))) < pi))∨((Abs(arg(t)) < pi/2)∧(2*Abs(arg(1 - x/(2*t))) <= pi))∨((Abs(arg(t)) < pi/2)∧(2*Abs(arg(1 - x/(2*t))) < pi)))), (Integral(exp(-|y| - (x - y)^2/(4*t)), (y, -oo, oo)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.