Sr Examen
Integral de x(1-x)^n dx
Solución
Ha introducido
[src]
1 / | | n | x*(1 - x) dx | / 0
$$\int\limits_{0}^{1} x \left(1 - x\right)^{n}\, dx$$
Integral(x*(1 - x)^n, (x, 0, 1))
Respuesta (Indefinida)
[src]
// 1 log(-1 + x) x*log(-1 + x) \
|| - ------ - ----------- + ------------- for n = -2|
/ || -1 + x -1 + x -1 + x |
| || |
| n || -x - log(-1 + x) for n = -1|
| x*(1 - x) dx = C + |< |
| || n 2 n 2 n n |
/ || (1 - x) x *(1 - x) n*x *(1 - x) n*x*(1 - x) |
||- ------------ + ------------ + ------------- - ------------ otherwise |
|| 2 2 2 2 |
\\ 2 + n + 3*n 2 + n + 3*n 2 + n + 3*n 2 + n + 3*n /
$$\int x \left(1 - x\right)^{n}\, dx = C + \begin{cases} \frac{x \log{\left(x - 1 \right)}}{x - 1} - \frac{\log{\left(x - 1 \right)}}{x - 1} - \frac{1}{x - 1} & \text{for}\: n = -2 \\- x - \log{\left(x - 1 \right)} & \text{for}\: n = -1 \\\frac{n x^{2} \left(1 - x\right)^{n}}{n^{2} + 3 n + 2} - \frac{n x \left(1 - x\right)^{n}}{n^{2} + 3 n + 2} + \frac{x^{2} \left(1 - x\right)^{n}}{n^{2} + 3 n + 2} - \frac{\left(1 - x\right)^{n}}{n^{2} + 3 n + 2} & \text{otherwise} \end{cases}$$
Respuesta
[src]
/ oo - pi*I for n = -2 | | oo + pi*I for n = -1 | < 1 |------------ otherwise | 2 |2 + n + 3*n \
$$\begin{cases} \infty - i \pi & \text{for}\: n = -2 \\\infty + i \pi & \text{for}\: n = -1 \\\frac{1}{n^{2} + 3 n + 2} & \text{otherwise} \end{cases}$$
=
=
/ oo - pi*I for n = -2 | | oo + pi*I for n = -1 | < 1 |------------ otherwise | 2 |2 + n + 3*n \
$$\begin{cases} \infty - i \pi & \text{for}\: n = -2 \\\infty + i \pi & \text{for}\: n = -1 \\\frac{1}{n^{2} + 3 n + 2} & \text{otherwise} \end{cases}$$
Piecewise((oo - pi*i, n = -2), (oo + pi*i, n = -1), (1/(2 + n^2 + 3*n), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.