Sr Examen
Integral de e^(ax^2) dx
Solución
Ha introducido
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oo / | | 2 | a*x | E dx | / 0
$$\int\limits_{0}^{\infty} e^{a x^{2}}\, dx$$
Integral(E^(a*x^2), (x, 0, oo))
Respuesta (Indefinida)
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/
| // ____ / ___\ \
| 2 ||-I*\/ pi *erf\I*x*\/ a / |
| a*x ||------------------------- for a != 0|
| E dx = C + |< ___ |
| || 2*\/ a |
/ || |
\\ x otherwise /
$$\int e^{a x^{2}}\, dx = C + \begin{cases} - \frac{i \sqrt{\pi} \operatorname{erf}{\left(i \sqrt{a} x \right)}}{2 \sqrt{a}} & \text{for}\: a \neq 0 \\x & \text{otherwise} \end{cases}$$
Respuesta
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/ ____ _______________ |-I*\/ pi *\/ polar_lift(a) pi |---------------------------- for |pi + arg(a)| <= -- | 2*a 2 | | oo | / < | | | 2 | | a*x | | e dx otherwise | | | / | 0 \
$$\begin{cases} - \frac{i \sqrt{\pi} \sqrt{\operatorname{polar\_lift}{\left(a \right)}}}{2 a} & \text{for}\: \left|{\arg{\left(a \right)} + \pi}\right| \leq \frac{\pi}{2} \\\int\limits_{0}^{\infty} e^{a x^{2}}\, dx & \text{otherwise} \end{cases}$$
=
=
/ ____ _______________ |-I*\/ pi *\/ polar_lift(a) pi |---------------------------- for |pi + arg(a)| <= -- | 2*a 2 | | oo | / < | | | 2 | | a*x | | e dx otherwise | | | / | 0 \
$$\begin{cases} - \frac{i \sqrt{\pi} \sqrt{\operatorname{polar\_lift}{\left(a \right)}}}{2 a} & \text{for}\: \left|{\arg{\left(a \right)} + \pi}\right| \leq \frac{\pi}{2} \\\int\limits_{0}^{\infty} e^{a x^{2}}\, dx & \text{otherwise} \end{cases}$$
Piecewise((-i*sqrt(pi)*sqrt(polar_lift(a))/(2*a), Abs(pi + arg(a)) <= pi/2), (Integral(exp(a*x^2), (x, 0, oo)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.