Sr Examen
Integral de exp(-m^2/(1+2/sqrt(6))x^(2+4/sqrt(6))) dx
Solución
Ha introducido
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oo / | | 4 | 2 + ----- | 2 ___ | -m \/ 6 | ---------*x | 2 | 1 + ----- | ___ | \/ 6 | e dx | / 0
$$\int\limits_{0}^{\infty} e^{x^{\frac{4}{\sqrt{6}} + 2} \frac{\left(-1\right) m^{2}}{\frac{2}{\sqrt{6}} + 1}}\, dx$$
Integral(exp(((-m^2)/(1 + 2/sqrt(6)))*x^(2 + 4/sqrt(6))), (x, 0, oo))
Respuesta (Indefinida)
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1
/ -------------
| / ___\ / ___\
| 4 | \/ 6 | | 2*\/ 6 |
| 2 + ----- 2*|1 + -----| | -------|
| 2 ___ \ 3 / | 2 2 3 |
| -m \/ 6 / ___\ / 1 \ | 1 m *x *x |
| ---------*x 3*\3 + \/ 6 / *Gamma|-------------|*lowergamma|-------------, --------------|
| 2 | / ___\| | / ___\ ___ |
| 1 + ----- | | \/ 6 || | | \/ 6 | \/ 6 |
| ___ |2*|1 + -----|| |2*|1 + -----| 1 + ----- |
| \/ 6 \ \ 3 // \ \ 3 / 3 /
| e dx = C + ------------------------------------------------------------------------------------------------------------------
| 1 1 1 1
/ ------------- --------- ------------- ---------
/ ___\ ___ / ___\ ___
| \/ 6 | \/ 6 | \/ 6 | \/ 6
2*|1 + -----| 1 + ----- 2*|1 + -----| 1 + -----
\ 3 / 3 / 3 \ \ 3 / ___ 3 / 3 \
20*3 *m *Gamma|1 + -------------| + 8*3 *\/ 6 *m *Gamma|1 + -------------|
| / ___\| | / ___\|
\ 2*\3 + \/ 6 // \ 2*\3 + \/ 6 //
$$\int e^{x^{\frac{4}{\sqrt{6}} + 2} \frac{\left(-1\right) m^{2}}{\frac{2}{\sqrt{6}} + 1}}\, dx = C + \frac{3 \left(\sqrt{6} + 3\right)^{\frac{1}{2 \left(\frac{\sqrt{6}}{3} + 1\right)}} \Gamma\left(\frac{1}{2 \left(\frac{\sqrt{6}}{3} + 1\right)}\right) \gamma\left(\frac{1}{2 \left(\frac{\sqrt{6}}{3} + 1\right)}, \frac{m^{2} x^{2} x^{\frac{2 \sqrt{6}}{3}}}{\frac{\sqrt{6}}{3} + 1}\right)}{8 \cdot 3^{\frac{1}{2 \left(\frac{\sqrt{6}}{3} + 1\right)}} \sqrt{6} m^{\frac{1}{\frac{\sqrt{6}}{3} + 1}} \Gamma\left(\frac{3}{2 \left(\sqrt{6} + 3\right)} + 1\right) + 20 \cdot 3^{\frac{1}{2 \left(\frac{\sqrt{6}}{3} + 1\right)}} m^{\frac{1}{\frac{\sqrt{6}}{3} + 1}} \Gamma\left(\frac{3}{2 \left(\sqrt{6} + 3\right)} + 1\right)}$$
Respuesta
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/ 1 | 1 - ----------- | ___ | 2*\/ 6 | 2 + ------- | 3 | / 2 \ 2 | |polar_lift (m)| / ___\ / 1 \ |2*|--------------| *\3 + \/ 6 / *Gamma|1 + -----------| | | ___ | | ___| | | \/ 6 | | 2*\/ 6 | | | 1 + ----- | | 2 + -------| | \ 3 / \ 3 / pi |--------------------------------------------------------------------- for 2*|arg(m)| <= -- | / ___\ 2 | 2 | 2*\/ 6 | | 9*m *|2 + -------| | \ 3 / < | oo | / | | | | ___ | | 2*\/ 6 | | 2 + ------- | | 2 3 | | -m *x | | ----------------- | | ___ | | \/ 6 | | 1 + ----- | | 3 | | e dx otherwise | | | / | 0 \
$$\begin{cases} \frac{2 \left(\frac{\operatorname{polar\_lift}^{2}{\left(m \right)}}{\frac{\sqrt{6}}{3} + 1}\right)^{1 - \frac{1}{\frac{2 \sqrt{6}}{3} + 2}} \left(\sqrt{6} + 3\right)^{2} \Gamma\left(\frac{1}{\frac{2 \sqrt{6}}{3} + 2} + 1\right)}{9 m^{2} \left(\frac{2 \sqrt{6}}{3} + 2\right)} & \text{for}\: 2 \left|{\arg{\left(m \right)}}\right| \leq \frac{\pi}{2} \\\int\limits_{0}^{\infty} e^{- \frac{m^{2} x^{\frac{2 \sqrt{6}}{3} + 2}}{\frac{\sqrt{6}}{3} + 1}}\, dx & \text{otherwise} \end{cases}$$
=
=
/ 1 | 1 - ----------- | ___ | 2*\/ 6 | 2 + ------- | 3 | / 2 \ 2 | |polar_lift (m)| / ___\ / 1 \ |2*|--------------| *\3 + \/ 6 / *Gamma|1 + -----------| | | ___ | | ___| | | \/ 6 | | 2*\/ 6 | | | 1 + ----- | | 2 + -------| | \ 3 / \ 3 / pi |--------------------------------------------------------------------- for 2*|arg(m)| <= -- | / ___\ 2 | 2 | 2*\/ 6 | | 9*m *|2 + -------| | \ 3 / < | oo | / | | | | ___ | | 2*\/ 6 | | 2 + ------- | | 2 3 | | -m *x | | ----------------- | | ___ | | \/ 6 | | 1 + ----- | | 3 | | e dx otherwise | | | / | 0 \
$$\begin{cases} \frac{2 \left(\frac{\operatorname{polar\_lift}^{2}{\left(m \right)}}{\frac{\sqrt{6}}{3} + 1}\right)^{1 - \frac{1}{\frac{2 \sqrt{6}}{3} + 2}} \left(\sqrt{6} + 3\right)^{2} \Gamma\left(\frac{1}{\frac{2 \sqrt{6}}{3} + 2} + 1\right)}{9 m^{2} \left(\frac{2 \sqrt{6}}{3} + 2\right)} & \text{for}\: 2 \left|{\arg{\left(m \right)}}\right| \leq \frac{\pi}{2} \\\int\limits_{0}^{\infty} e^{- \frac{m^{2} x^{\frac{2 \sqrt{6}}{3} + 2}}{\frac{\sqrt{6}}{3} + 1}}\, dx & \text{otherwise} \end{cases}$$
Piecewise((2*(polar_lift(m)^2/(1 + sqrt(6)/3))^(1 - 1/(2 + 2*sqrt(6)/3))*(3 + sqrt(6))^2*gamma(1 + 1/(2 + 2*sqrt(6)/3))/(9*m^2*(2 + 2*sqrt(6)/3)), 2*Abs(arg(m)) <= pi/2), (Integral(exp(-m^2*x^(2 + 2*sqrt(6)/3)/(1 + sqrt(6)/3)), (x, 0, oo)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.