Sr Examen
Integral de exp((-x^2)/(a^2)+(b*x)/(a^2)) dx
Solución
Ha introducido
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oo / | | 2 | -x b*x | ---- + --- | 2 2 | a a | e dx | / -oo
$$\int\limits_{-\infty}^{\infty} e^{\frac{\left(-1\right) x^{2}}{a^{2}} + \frac{b x}{a^{2}}}\, dx$$
Integral(exp((-x^2)/a^2 + (b*x)/a^2), (x, -oo, oo))
Solución detallada
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Ahora simplificar:
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Añadimos la constante de integración:
ErfRule(a=-1/a**2, b=b/a**2, c=0, context=exp((-x**2)/a**2 + (b*x)/a**2), symbol=x)
Respuesta:
Respuesta (Indefinida)
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2
/ b 2*x \ b
| -- - --- | ----
_____ | 2 2 | 2
/ ____ / 2 | a a | 4*a
| \/ pi *\/ -a *erfi|------------|*e
| 2 | _____|
| -x b*x | / -1 |
| ---- + --- |2* / --- |
| 2 2 | / 2 |
| a a \ \/ a /
| e dx = C + ----------------------------------------
| 2
/
$$\int e^{\frac{\left(-1\right) x^{2}}{a^{2}} + \frac{b x}{a^{2}}}\, dx = C + \frac{\sqrt{\pi} \sqrt{- a^{2}} e^{\frac{b^{2}}{4 a^{2}}} \operatorname{erfi}{\left(\frac{\frac{b}{a^{2}} - \frac{2 x}{a^{2}}}{2 \sqrt{- \frac{1}{a^{2}}}} \right)}}{2}$$
Respuesta
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/ 2 2 | b b | ---- ---- | 2 2 | ____ / / b \\ 4*a ____ / b \ 4*a |a*\/ pi *|2 - erfc|---||*e a*\/ pi *erfc|---|*e | \ \2*a// \2*a/ / / / pi \ / pi\ / pi\ pi\ / / pi \ / pi\ / pi\ pi\\ |------------------------------ + ------------------------ for And|Or|And|2*|arg(a)| <= --, |-2*arg(b) + 4*arg(a)| < pi|, And||-2*arg(b) + 4*arg(a)| <= pi, 2*|arg(a)| < --|, And||-2*arg(b) + 4*arg(a)| < pi, 2*|arg(a)| < --|, 2*|arg(a)| < --|, Or|And|2*|arg(a)| <= --, |-4*arg(a) + 2*pi + 2*arg(b)| < pi|, And||-4*arg(a) + 2*pi + 2*arg(b)| <= pi, 2*|arg(a)| < --|, And||-4*arg(a) + 2*pi + 2*arg(b)| < pi, 2*|arg(a)| < --|, 2*|arg(a)| < --|| | 2 2 \ \ \ 2 / \ 2 / \ 2 / 2 / \ \ 2 / \ 2 / \ 2 / 2 // | | oo < / | | | | 2 | | x b*x | | - -- + --- | | 2 2 | | a a | | e dx otherwise | | | / | -oo \
$$\begin{cases} \frac{\sqrt{\pi} a \left(2 - \operatorname{erfc}{\left(\frac{b}{2 a} \right)}\right) e^{\frac{b^{2}}{4 a^{2}}}}{2} + \frac{\sqrt{\pi} a e^{\frac{b^{2}}{4 a^{2}}} \operatorname{erfc}{\left(\frac{b}{2 a} \right)}}{2} & \text{for}\: \left(\left(2 \left|{\arg{\left(a \right)}}\right| \leq \frac{\pi}{2} \wedge \left|{4 \arg{\left(a \right)} - 2 \arg{\left(b \right)}}\right| < \pi\right) \vee \left(\left|{4 \arg{\left(a \right)} - 2 \arg{\left(b \right)}}\right| \leq \pi \wedge 2 \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2}\right) \vee \left(\left|{4 \arg{\left(a \right)} - 2 \arg{\left(b \right)}}\right| < \pi \wedge 2 \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2}\right) \vee 2 \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2}\right) \wedge \left(\left(2 \left|{\arg{\left(a \right)}}\right| \leq \frac{\pi}{2} \wedge \left|{- 4 \arg{\left(a \right)} + 2 \arg{\left(b \right)} + 2 \pi}\right| < \pi\right) \vee \left(\left|{- 4 \arg{\left(a \right)} + 2 \arg{\left(b \right)} + 2 \pi}\right| \leq \pi \wedge 2 \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2}\right) \vee \left(\left|{- 4 \arg{\left(a \right)} + 2 \arg{\left(b \right)} + 2 \pi}\right| < \pi \wedge 2 \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2}\right) \vee 2 \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2}\right) \\\int\limits_{-\infty}^{\infty} e^{\frac{b x}{a^{2}} - \frac{x^{2}}{a^{2}}}\, dx & \text{otherwise} \end{cases}$$
=
=
/ 2 2 | b b | ---- ---- | 2 2 | ____ / / b \\ 4*a ____ / b \ 4*a |a*\/ pi *|2 - erfc|---||*e a*\/ pi *erfc|---|*e | \ \2*a// \2*a/ / / / pi \ / pi\ / pi\ pi\ / / pi \ / pi\ / pi\ pi\\ |------------------------------ + ------------------------ for And|Or|And|2*|arg(a)| <= --, |-2*arg(b) + 4*arg(a)| < pi|, And||-2*arg(b) + 4*arg(a)| <= pi, 2*|arg(a)| < --|, And||-2*arg(b) + 4*arg(a)| < pi, 2*|arg(a)| < --|, 2*|arg(a)| < --|, Or|And|2*|arg(a)| <= --, |-4*arg(a) + 2*pi + 2*arg(b)| < pi|, And||-4*arg(a) + 2*pi + 2*arg(b)| <= pi, 2*|arg(a)| < --|, And||-4*arg(a) + 2*pi + 2*arg(b)| < pi, 2*|arg(a)| < --|, 2*|arg(a)| < --|| | 2 2 \ \ \ 2 / \ 2 / \ 2 / 2 / \ \ 2 / \ 2 / \ 2 / 2 // | | oo < / | | | | 2 | | x b*x | | - -- + --- | | 2 2 | | a a | | e dx otherwise | | | / | -oo \
$$\begin{cases} \frac{\sqrt{\pi} a \left(2 - \operatorname{erfc}{\left(\frac{b}{2 a} \right)}\right) e^{\frac{b^{2}}{4 a^{2}}}}{2} + \frac{\sqrt{\pi} a e^{\frac{b^{2}}{4 a^{2}}} \operatorname{erfc}{\left(\frac{b}{2 a} \right)}}{2} & \text{for}\: \left(\left(2 \left|{\arg{\left(a \right)}}\right| \leq \frac{\pi}{2} \wedge \left|{4 \arg{\left(a \right)} - 2 \arg{\left(b \right)}}\right| < \pi\right) \vee \left(\left|{4 \arg{\left(a \right)} - 2 \arg{\left(b \right)}}\right| \leq \pi \wedge 2 \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2}\right) \vee \left(\left|{4 \arg{\left(a \right)} - 2 \arg{\left(b \right)}}\right| < \pi \wedge 2 \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2}\right) \vee 2 \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2}\right) \wedge \left(\left(2 \left|{\arg{\left(a \right)}}\right| \leq \frac{\pi}{2} \wedge \left|{- 4 \arg{\left(a \right)} + 2 \arg{\left(b \right)} + 2 \pi}\right| < \pi\right) \vee \left(\left|{- 4 \arg{\left(a \right)} + 2 \arg{\left(b \right)} + 2 \pi}\right| \leq \pi \wedge 2 \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2}\right) \vee \left(\left|{- 4 \arg{\left(a \right)} + 2 \arg{\left(b \right)} + 2 \pi}\right| < \pi \wedge 2 \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2}\right) \vee 2 \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2}\right) \\\int\limits_{-\infty}^{\infty} e^{\frac{b x}{a^{2}} - \frac{x^{2}}{a^{2}}}\, dx & \text{otherwise} \end{cases}$$
Piecewise((a*sqrt(pi)*(2 - erfc(b/(2*a)))*exp(b^2/(4*a^2))/2 + a*sqrt(pi)*erfc(b/(2*a))*exp(b^2/(4*a^2))/2, ((2*Abs(arg(a)) < pi/2)∨((2*Abs(arg(a)) <= pi/2)∧(Abs(-2*arg(b) + 4*arg(a)) < pi))∨((2*Abs(arg(a)) < pi/2)∧(Abs(-2*arg(b) + 4*arg(a)) <= pi))∨((2*Abs(arg(a)) < pi/2)∧(Abs(-2*arg(b) + 4*arg(a)) < pi)))∧((2*Abs(arg(a)) < pi/2)∨((2*Abs(arg(a)) <= pi/2)∧(Abs(-4*arg(a) + 2*pi + 2*arg(b)) < pi))∨((2*Abs(arg(a)) < pi/2)∧(Abs(-4*arg(a) + 2*pi + 2*arg(b)) <= pi))∨((2*Abs(arg(a)) < pi/2)∧(Abs(-4*arg(a) + 2*pi + 2*arg(b)) < pi)))), (Integral(exp(-x^2/a^2 + b*x/a^2), (x, -oo, oo)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.