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