Sr Examen
Integral de (x^2)*exp(-2*a*x) dx
Solución
Ha introducido
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1 / | | 2 -2*a*x | x *e dx | / 0
$$\int\limits_{0}^{1} x^{2} e^{- 2 a x}\, dx$$
Integral(x^2*exp((-2*a)*x), (x, 0, 1))
Respuesta (Indefinida)
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// 3 \
|| x |
|| -- for a = 0|
|| 3 |
|| |
/ ||/ -2*a*x | // x for a = 0\
| |||(1 + 2*a*x)*e 3 | || |
| 2 -2*a*x |||------------------- for 8*a != 0 | 2 || -2*a*x |
| x *e dx = C - 2*|<| 3 | + x *|<-e |
| ||| 8*a | ||--------- otherwise|
/ ||< otherwise| || 2*a |
||| 2 | \\ /
||| -x |
||| ---- otherwise |
||| 4*a |
||\ |
\\ /
$$\int x^{2} e^{- 2 a x}\, dx = C + x^{2} \left(\begin{cases} x & \text{for}\: a = 0 \\- \frac{e^{- 2 a x}}{2 a} & \text{otherwise} \end{cases}\right) - 2 \left(\begin{cases} \frac{x^{3}}{3} & \text{for}\: a = 0 \\\begin{cases} \frac{\left(2 a x + 1\right) e^{- 2 a x}}{8 a^{3}} & \text{for}\: 8 a^{3} \neq 0 \\- \frac{x^{2}}{4 a} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
Respuesta
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/ / 2\ -2*a | 1 \-1 - 2*a - 2*a /*e |---- + ----------------------- for And(a > -oo, a < oo, a != 0) < 3 3 |4*a 4*a | \ 1/3 otherwise
$$\begin{cases} \frac{\left(- 2 a^{2} - 2 a - 1\right) e^{- 2 a}}{4 a^{3}} + \frac{1}{4 a^{3}} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\\frac{1}{3} & \text{otherwise} \end{cases}$$
=
=
/ / 2\ -2*a | 1 \-1 - 2*a - 2*a /*e |---- + ----------------------- for And(a > -oo, a < oo, a != 0) < 3 3 |4*a 4*a | \ 1/3 otherwise
$$\begin{cases} \frac{\left(- 2 a^{2} - 2 a - 1\right) e^{- 2 a}}{4 a^{3}} + \frac{1}{4 a^{3}} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\\frac{1}{3} & \text{otherwise} \end{cases}$$
Piecewise((1/(4*a^3) + (-1 - 2*a - 2*a^2)*exp(-2*a)/(4*a^3), (a > -oo)∧(a < oo)∧(Ne(a, 0))), (1/3, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.