Sr Examen
Integral de sqrt(x)*e^(-x/(2r^2)) dx
Solución
Ha introducido
[src]
oo / | | -x | ---- | 2 | ___ 2*r | \/ x *E dx | / 0
$$\int\limits_{0}^{\infty} e^{\frac{\left(-1\right) x}{2 r^{2}}} \sqrt{x}\, dx$$
Integral(sqrt(x)*E^((-x)/((2*r^2))), (x, 0, oo))
Solución detallada
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Ahora simplificar:
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Añadimos la constante de integración:
UpperGammaRule(a=-1/(2*r**2), e=1/2, context=E**((-x)/((2*r**2)))*sqrt(x), symbol=x)
Respuesta:
Respuesta (Indefinida)
[src]
/ / ____\ -x \
| | ___ / x | ----|
| |\/ 2 * / -- | ____ 2|
/ | | / 2 | ___ / x 2*r |
| | ____ | \/ r | \/ 2 * / -- *e |
| -x |\/ pi *erfc|---------------| / 2 |
| ---- ___ 2 ___ | \ 2 / \/ r |
| 2 2*\/ 2 *r *\/ x *|---------------------------- + ---------------------|
| ___ 2*r \ 2 2 /
| \/ x *E dx = C - -----------------------------------------------------------------------
| ____
/ / x
/ --
/ 2
\/ r
$$\int e^{\frac{\left(-1\right) x}{2 r^{2}}} \sqrt{x}\, dx = C - \frac{2 \sqrt{2} r^{2} \sqrt{x} \left(\frac{\sqrt{2} \sqrt{\frac{x}{r^{2}}} e^{- \frac{x}{2 r^{2}}}}{2} + \frac{\sqrt{\pi} \operatorname{erfc}{\left(\frac{\sqrt{2} \sqrt{\frac{x}{r^{2}}}}{2} \right)}}{2}\right)}{\sqrt{\frac{x}{r^{2}}}}$$
Respuesta
[src]
/ ___ ____ 3 pi | \/ 2 *\/ pi *r for 2*|arg(r)| < -- | 2 | | oo | / | | < | -x | | ---- | | 2 | | ___ 2*r | | \/ x *e dx otherwise | | |/ \0
$$\begin{cases} \sqrt{2} \sqrt{\pi} r^{3} & \text{for}\: 2 \left|{\arg{\left(r \right)}}\right| < \frac{\pi}{2} \\\int\limits_{0}^{\infty} \sqrt{x} e^{- \frac{x}{2 r^{2}}}\, dx & \text{otherwise} \end{cases}$$
=
=
/ ___ ____ 3 pi | \/ 2 *\/ pi *r for 2*|arg(r)| < -- | 2 | | oo | / | | < | -x | | ---- | | 2 | | ___ 2*r | | \/ x *e dx otherwise | | |/ \0
$$\begin{cases} \sqrt{2} \sqrt{\pi} r^{3} & \text{for}\: 2 \left|{\arg{\left(r \right)}}\right| < \frac{\pi}{2} \\\int\limits_{0}^{\infty} \sqrt{x} e^{- \frac{x}{2 r^{2}}}\, dx & \text{otherwise} \end{cases}$$
Piecewise((sqrt(2)*sqrt(pi)*r^3, 2*Abs(arg(r)) < pi/2), (Integral(sqrt(x)*exp(-x/(2*r^2)), (x, 0, oo)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.