Sr Examen
Integral de |x-1|^n dx
Solución
Ha introducido
[src]
2 / | | n | |x - 1| dx | / -2
$$\int\limits_{-2}^{2} \left|{x - 1}\right|^{n}\, dx$$
Integral(|x - 1|^n, (x, -2, 2))
Respuesta (Indefinida)
[src]
// n /pi*n\ n /pi*n\ \
|| 2*(1 - x) *Gamma(-n)*Gamma(1 + n)*sin|----| 2*x*(1 - x) *Gamma(-n)*Gamma(1 + n)*sin|----| |
|| \ 2 / \ 2 / |
||------------------------------------------------------- - ------------------------------------------------------- for |x| < 1|
/ || /1 n\ /1 n\ /1 n\ /1 n\ /1 n\ /1 n\ /1 n\ /1 n\ |
| ||Gamma|- + -|*Gamma|- - -| + n*Gamma|- + -|*Gamma|- - -| Gamma|- + -|*Gamma|- - -| + n*Gamma|- + -|*Gamma|- - -| |
| n || \2 2/ \2 2/ \2 2/ \2 2/ \2 2/ \2 2/ \2 2/ \2 2/ |
| |x - 1| dx = C + |< |
| || / 3 n | \ |
/ || |2 + n, 1 - + - | | |
|| __1, 2 | 2 2 | | /pi*n\ |
|| -2*Gamma(1 + n)*/__ | | x|*sin|----| otherwise |
|| \_|3, 3 | 3 n | | \ 2 / |
|| | 1 - + -, 0 | | |
\\ \ 2 2 | / /
$$\int \left|{x - 1}\right|^{n}\, dx = C + \begin{cases} - \frac{2 x \left(1 - x\right)^{n} \sin{\left(\frac{\pi n}{2} \right)} \Gamma\left(- n\right) \Gamma\left(n + 1\right)}{n \Gamma\left(\frac{1}{2} - \frac{n}{2}\right) \Gamma\left(\frac{n}{2} + \frac{1}{2}\right) + \Gamma\left(\frac{1}{2} - \frac{n}{2}\right) \Gamma\left(\frac{n}{2} + \frac{1}{2}\right)} + \frac{2 \left(1 - x\right)^{n} \sin{\left(\frac{\pi n}{2} \right)} \Gamma\left(- n\right) \Gamma\left(n + 1\right)}{n \Gamma\left(\frac{1}{2} - \frac{n}{2}\right) \Gamma\left(\frac{n}{2} + \frac{1}{2}\right) + \Gamma\left(\frac{1}{2} - \frac{n}{2}\right) \Gamma\left(\frac{n}{2} + \frac{1}{2}\right)} & \text{for}\: \left|{x}\right| < 1 \\- 2 {G_{3, 3}^{1, 2}\left(\begin{matrix} n + 2, 1 & \frac{n}{2} + \frac{3}{2} \\1 & \frac{n}{2} + \frac{3}{2}, 0 \end{matrix} \middle| {x} \right)} \sin{\left(\frac{\pi n}{2} \right)} \Gamma\left(n + 1\right) & \text{otherwise} \end{cases}$$
Respuesta
[src]
/ 1 + n 1 + n | 1 3 2*0 |----- + ------ - -------- for And(n > -oo, n < oo, n != -1) <1 + n 1 + n 1 + n | | oo otherwise \
$$\begin{cases} - \frac{2 \cdot 0^{n + 1}}{n + 1} + \frac{3^{n + 1}}{n + 1} + \frac{1}{n + 1} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq -1 \\\infty & \text{otherwise} \end{cases}$$
=
=
/ 1 + n 1 + n | 1 3 2*0 |----- + ------ - -------- for And(n > -oo, n < oo, n != -1) <1 + n 1 + n 1 + n | | oo otherwise \
$$\begin{cases} - \frac{2 \cdot 0^{n + 1}}{n + 1} + \frac{3^{n + 1}}{n + 1} + \frac{1}{n + 1} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq -1 \\\infty & \text{otherwise} \end{cases}$$
Piecewise((1/(1 + n) + 3^(1 + n)/(1 + n) - 2*0^(1 + n)/(1 + n), (n > -oo)∧(n < oo)∧(Ne(n, -1))), (oo, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.