Sr Examen
Integral de exp(a*x-b*x^2) dx
Solución
Ha introducido
[src]
y / | | 2 | a*x - b*x | e dx | / 0
$$\int\limits_{0}^{y} e^{a x - b x^{2}}\, dx$$
Integral(exp(a*x - b*x^2), (x, 0, y))
Solución detallada
-
Añadimos la constante de integración:
ErfRule(a=-b, b=a, c=0, context=exp(a*x - b*x**2), symbol=x)
Respuesta:
Respuesta (Indefinida)
[src]
2
a
_____ ---
/ ____ / -1 /a - 2*b*x\ 4*b
| \/ pi * / --- *erfi|---------|*e
| 2 \/ b | ____|
| a*x - b*x \ 2*\/ -b /
| e dx = C + -------------------------------------
| 2
/
$$\int e^{a x - b x^{2}}\, dx = C + \frac{\sqrt{\pi} \sqrt{- \frac{1}{b}} e^{\frac{a^{2}}{4 b}} \operatorname{erfi}{\left(\frac{a - 2 b x}{2 \sqrt{- b}} \right)}}{2}$$
Respuesta
[src]
2 2
a a
_____ --- _____ ---
____ / -1 /a - 2*b*y\ 4*b ____ / -1 / a \ 4*b
\/ pi * / --- *erfi|---------|*e \/ pi * / --- *erfi|--------|*e
\/ b | ____| \/ b | ____|
\ 2*\/ -b / \2*\/ -b /
------------------------------------- - ------------------------------------
2 2
$$- \frac{\sqrt{\pi} \sqrt{- \frac{1}{b}} e^{\frac{a^{2}}{4 b}} \operatorname{erfi}{\left(\frac{a}{2 \sqrt{- b}} \right)}}{2} + \frac{\sqrt{\pi} \sqrt{- \frac{1}{b}} e^{\frac{a^{2}}{4 b}} \operatorname{erfi}{\left(\frac{a - 2 b y}{2 \sqrt{- b}} \right)}}{2}$$
=
=
2 2
a a
_____ --- _____ ---
____ / -1 /a - 2*b*y\ 4*b ____ / -1 / a \ 4*b
\/ pi * / --- *erfi|---------|*e \/ pi * / --- *erfi|--------|*e
\/ b | ____| \/ b | ____|
\ 2*\/ -b / \2*\/ -b /
------------------------------------- - ------------------------------------
2 2
$$- \frac{\sqrt{\pi} \sqrt{- \frac{1}{b}} e^{\frac{a^{2}}{4 b}} \operatorname{erfi}{\left(\frac{a}{2 \sqrt{- b}} \right)}}{2} + \frac{\sqrt{\pi} \sqrt{- \frac{1}{b}} e^{\frac{a^{2}}{4 b}} \operatorname{erfi}{\left(\frac{a - 2 b y}{2 \sqrt{- b}} \right)}}{2}$$
sqrt(pi)*sqrt(-1/b)*erfi((a - 2*b*y)/(2*sqrt(-b)))*exp(a^2/(4*b))/2 - sqrt(pi)*sqrt(-1/b)*erfi(a/(2*sqrt(-b)))*exp(a^2/(4*b))/2
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.