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