Sr Examen
Integral de exp(p*(-i)*x)*a/(a^2+x^2) dx
Solución
Ha introducido
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oo / | | p*(-I)*x | e *a | ----------- dx | 2 2 | a + x | / -oo
$$\int\limits_{-\infty}^{\infty} \frac{a e^{x - i p}}{a^{2} + x^{2}}\, dx$$
Integral((exp((p*(-i))*x)*a)/(a^2 + x^2), (x, -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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/ / | | | p*(-I)*x | -I*p*x | e *a | e | ----------- dx = C + a* | ------- dx | 2 2 | 2 2 | a + x | a + x | | / /
$$\int \frac{a e^{x - i p}}{a^{2} + x^{2}}\, dx = C + a \int \frac{e^{- i p x}}{a^{2} + x^{2}}\, dx$$
Respuesta
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/ / pi \ / pi \ /pi*I \ / pi*I \ |- |- -- + I*Shi(a*p)|*cosh(a*p) - |- -- - I*Shi(a*p)|*cosh(a*p) + I*|---- + Chi(a*p)|*sinh(a*p) - I*|- ---- + Chi(a*p)|*sinh(a*p) for And(Or(And(Or(And(2*|arg(a)| < pi, 2*|arg(a)| != pi), 2*|arg(a)| < pi), |-pi + 2*arg(p)| < pi), And(|-pi + 2*arg(p)| = pi, 2*|arg(a)| < pi), And(|-pi + 2*arg(p)| < pi, 2*|arg(a)| < pi)), Or(And(Or(And(2*|arg(a)| < pi, 2*|arg(a)| != pi), 2*|arg(a)| < pi), |pi + 2*arg(p)| < pi), And(|pi + 2*arg(p)| = pi, 2*|arg(a)| < pi), And(|pi + 2*arg(p)| < pi, 2*|arg(a)| < pi))) | \ 2 / \ 2 / \ 2 / \ 2 / | | oo | / | | < | -I*p*x | | a*e | | --------- dx otherwise | | 2 2 | | a + x | | | / \ -oo
$$\begin{cases} - \left(- i \operatorname{Shi}{\left(a p \right)} - \frac{\pi}{2}\right) \cosh{\left(a p \right)} - \left(i \operatorname{Shi}{\left(a p \right)} - \frac{\pi}{2}\right) \cosh{\left(a p \right)} - i \left(\operatorname{Chi}\left(a p\right) - \frac{i \pi}{2}\right) \sinh{\left(a p \right)} + i \left(\operatorname{Chi}\left(a p\right) + \frac{i \pi}{2}\right) \sinh{\left(a p \right)} & \text{for}\: \left(\left(\left(\left(2 \left|{\arg{\left(a \right)}}\right| < \pi \wedge 2 \left|{\arg{\left(a \right)}}\right| \neq \pi\right) \vee 2 \left|{\arg{\left(a \right)}}\right| < \pi\right) \wedge \left|{2 \arg{\left(p \right)} - \pi}\right| < \pi\right) \vee \left(\left|{2 \arg{\left(p \right)} - \pi}\right| = \pi \wedge 2 \left|{\arg{\left(a \right)}}\right| < \pi\right) \vee \left(\left|{2 \arg{\left(p \right)} - \pi}\right| < \pi \wedge 2 \left|{\arg{\left(a \right)}}\right| < \pi\right)\right) \wedge \left(\left(\left(\left(2 \left|{\arg{\left(a \right)}}\right| < \pi \wedge 2 \left|{\arg{\left(a \right)}}\right| \neq \pi\right) \vee 2 \left|{\arg{\left(a \right)}}\right| < \pi\right) \wedge \left|{2 \arg{\left(p \right)} + \pi}\right| < \pi\right) \vee \left(\left|{2 \arg{\left(p \right)} + \pi}\right| = \pi \wedge 2 \left|{\arg{\left(a \right)}}\right| < \pi\right) \vee \left(\left|{2 \arg{\left(p \right)} + \pi}\right| < \pi \wedge 2 \left|{\arg{\left(a \right)}}\right| < \pi\right)\right) \\\int\limits_{-\infty}^{\infty} \frac{a e^{- i p x}}{a^{2} + x^{2}}\, dx & \text{otherwise} \end{cases}$$
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/ / pi \ / pi \ /pi*I \ / pi*I \ |- |- -- + I*Shi(a*p)|*cosh(a*p) - |- -- - I*Shi(a*p)|*cosh(a*p) + I*|---- + Chi(a*p)|*sinh(a*p) - I*|- ---- + Chi(a*p)|*sinh(a*p) for And(Or(And(Or(And(2*|arg(a)| < pi, 2*|arg(a)| != pi), 2*|arg(a)| < pi), |-pi + 2*arg(p)| < pi), And(|-pi + 2*arg(p)| = pi, 2*|arg(a)| < pi), And(|-pi + 2*arg(p)| < pi, 2*|arg(a)| < pi)), Or(And(Or(And(2*|arg(a)| < pi, 2*|arg(a)| != pi), 2*|arg(a)| < pi), |pi + 2*arg(p)| < pi), And(|pi + 2*arg(p)| = pi, 2*|arg(a)| < pi), And(|pi + 2*arg(p)| < pi, 2*|arg(a)| < pi))) | \ 2 / \ 2 / \ 2 / \ 2 / | | oo | / | | < | -I*p*x | | a*e | | --------- dx otherwise | | 2 2 | | a + x | | | / \ -oo
$$\begin{cases} - \left(- i \operatorname{Shi}{\left(a p \right)} - \frac{\pi}{2}\right) \cosh{\left(a p \right)} - \left(i \operatorname{Shi}{\left(a p \right)} - \frac{\pi}{2}\right) \cosh{\left(a p \right)} - i \left(\operatorname{Chi}\left(a p\right) - \frac{i \pi}{2}\right) \sinh{\left(a p \right)} + i \left(\operatorname{Chi}\left(a p\right) + \frac{i \pi}{2}\right) \sinh{\left(a p \right)} & \text{for}\: \left(\left(\left(\left(2 \left|{\arg{\left(a \right)}}\right| < \pi \wedge 2 \left|{\arg{\left(a \right)}}\right| \neq \pi\right) \vee 2 \left|{\arg{\left(a \right)}}\right| < \pi\right) \wedge \left|{2 \arg{\left(p \right)} - \pi}\right| < \pi\right) \vee \left(\left|{2 \arg{\left(p \right)} - \pi}\right| = \pi \wedge 2 \left|{\arg{\left(a \right)}}\right| < \pi\right) \vee \left(\left|{2 \arg{\left(p \right)} - \pi}\right| < \pi \wedge 2 \left|{\arg{\left(a \right)}}\right| < \pi\right)\right) \wedge \left(\left(\left(\left(2 \left|{\arg{\left(a \right)}}\right| < \pi \wedge 2 \left|{\arg{\left(a \right)}}\right| \neq \pi\right) \vee 2 \left|{\arg{\left(a \right)}}\right| < \pi\right) \wedge \left|{2 \arg{\left(p \right)} + \pi}\right| < \pi\right) \vee \left(\left|{2 \arg{\left(p \right)} + \pi}\right| = \pi \wedge 2 \left|{\arg{\left(a \right)}}\right| < \pi\right) \vee \left(\left|{2 \arg{\left(p \right)} + \pi}\right| < \pi \wedge 2 \left|{\arg{\left(a \right)}}\right| < \pi\right)\right) \\\int\limits_{-\infty}^{\infty} \frac{a e^{- i p x}}{a^{2} + x^{2}}\, dx & \text{otherwise} \end{cases}$$
Piecewise((-(-pi/2 + i*Shi(a*p))*cosh(a*p) - (-pi/2 - i*Shi(a*p))*cosh(a*p) + i*(pi*i/2 + Chi(a*p))*sinh(a*p) - i*(-pi*i/2 + Chi(a*p))*sinh(a*p), (((2*Abs(arg(a)) < pi)∧(Abs(pi + 2*arg(p)) = pi)))∨((2*Abs(arg(a)) < pi)∧(Abs(pi + 2*arg(p)) < pi))∨((Abs(pi + 2*arg(p)) < pi)∧((2*Abs(arg(a)) < pi)∨((2*Abs(arg(a)) < pi)∧(Ne(2*Abs(arg(a), pi))))))∧(((2*Abs(arg(a)) < pi)∧(Abs(-pi + 2*arg(p)) = pi)))∨((2*Abs(arg(a)) < pi)∧(Abs(-pi + 2*arg(p)) < pi))∨((Abs(-pi + 2*arg(p)) < pi)∧((2*Abs(arg(a)) < pi)∨((2*Abs(arg(a)) < pi)∧(Ne(2*Abs(arg(a), pi))))))), (Integral(a*exp(-i*p*x)/(a^2 + x^2), (x, -oo, oo)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.