Sr Examen
Integral de (x^4)*exp(-2*a*x) dx
Solución
Ha introducido
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1 / | | 4 -2*a*x | x *e dx | / 0
$$\int\limits_{0}^{1} x^{4} e^{- 2 a x}\, dx$$
Integral(x^4*exp((-2*a)*x), (x, 0, 1))
Respuesta (Indefinida)
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// 5 \
|| x |
|| -- for a = 0|
|| 5 |
|| |
/ ||// 3 3 2 2\ -2*a*x | // x for a = 0\
| |||\3 + 4*a *x + 6*a*x + 6*a *x /*e 5 | || |
| 4 -2*a*x |||--------------------------------------- for 16*a != 0 | 4 || -2*a*x |
| x *e dx = C - 4*|<| 5 | + x *|<-e |
| ||| 16*a | ||--------- otherwise|
/ ||< otherwise| || 2*a |
||| 4 | \\ /
||| -x |
||| ---- otherwise |
||| 8*a |
||\ |
\\ /
$$\int x^{4} e^{- 2 a x}\, dx = C + x^{4} \left(\begin{cases} x & \text{for}\: a = 0 \\- \frac{e^{- 2 a x}}{2 a} & \text{otherwise} \end{cases}\right) - 4 \left(\begin{cases} \frac{x^{5}}{5} & \text{for}\: a = 0 \\\begin{cases} \frac{\left(4 a^{3} x^{3} + 6 a^{2} x^{2} + 6 a x + 3\right) e^{- 2 a x}}{16 a^{5}} & \text{for}\: 16 a^{5} \neq 0 \\- \frac{x^{4}}{8 a} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
Respuesta
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/ / 2 3 4\ -2*a | 3 \-3 - 6*a - 6*a - 4*a - 2*a /*e |---- + ------------------------------------- for And(a > -oo, a < oo, a != 0) < 5 5 |4*a 4*a | \ 1/5 otherwise
$$\begin{cases} \frac{\left(- 2 a^{4} - 4 a^{3} - 6 a^{2} - 6 a - 3\right) e^{- 2 a}}{4 a^{5}} + \frac{3}{4 a^{5}} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\\frac{1}{5} & \text{otherwise} \end{cases}$$
=
=
/ / 2 3 4\ -2*a | 3 \-3 - 6*a - 6*a - 4*a - 2*a /*e |---- + ------------------------------------- for And(a > -oo, a < oo, a != 0) < 5 5 |4*a 4*a | \ 1/5 otherwise
$$\begin{cases} \frac{\left(- 2 a^{4} - 4 a^{3} - 6 a^{2} - 6 a - 3\right) e^{- 2 a}}{4 a^{5}} + \frac{3}{4 a^{5}} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\\frac{1}{5} & \text{otherwise} \end{cases}$$
Piecewise((3/(4*a^5) + (-3 - 6*a - 6*a^2 - 4*a^3 - 2*a^4)*exp(-2*a)/(4*a^5), (a > -oo)∧(a < oo)∧(Ne(a, 0))), (1/5, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.