Sr Examen
Integral de x^2÷(1-x)^n dx
Solución
Ha introducido
[src]
1 / | | 2 | x | -------- dx | n | (1 - x) | / 0
$$\int\limits_{0}^{1} \frac{x^{2}}{\left(1 - x\right)^{n}}\, dx$$
Integral(x^2/(1 - x)^n, (x, 0, 1))
Respuesta (Indefinida)
[src]
// 2 \
|| x |
|| -x - log(-1 + x) - -- for n = 1|
|| 2 |
|| |
|| 2 |
/ || 2 x 2*log(-1 + x) 2*x*log(-1 + x) |
| || - ------ + ------ - ------------- + --------------- for n = 2|
| 2 || -1 + x -1 + x -1 + x -1 + x |
| x || |
| -------- dx = C + |< 2 |
| n || 3 2*log(-1 + x) 4*x 2*x *log(-1 + x) 4*x*log(-1 + x) |
| (1 - x) || - -------------- - -------------- + -------------- - ---------------- + --------------- for n = 3|
| || 2 2 2 2 2 |
/ || 2 - 4*x + 2*x 2 - 4*x + 2*x 2 - 4*x + 2*x 2 - 4*x + 2*x 2 - 4*x + 2*x |
|| |
|| 3 2 2 2 2 3 3 |
|| 2 2*x n *x n*x n *x 2*n*x 3*n*x |
||---------------------------------------------------------- - ---------------------------------------------------------- + ---------------------------------------------------------- - ---------------------------------------------------------- - ---------------------------------------------------------- - ---------------------------------------------------------- + ---------------------------------------------------------- otherwise|
|| n 3 n 2 n n n 3 n 2 n n n 3 n 2 n n n 3 n 2 n n n 3 n 2 n n n 3 n 2 n n n 3 n 2 n n |
\\- 6*(1 - x) + n *(1 - x) - 6*n *(1 - x) + 11*n*(1 - x) - 6*(1 - x) + n *(1 - x) - 6*n *(1 - x) + 11*n*(1 - x) - 6*(1 - x) + n *(1 - x) - 6*n *(1 - x) + 11*n*(1 - x) - 6*(1 - x) + n *(1 - x) - 6*n *(1 - x) + 11*n*(1 - x) - 6*(1 - x) + n *(1 - x) - 6*n *(1 - x) + 11*n*(1 - x) - 6*(1 - x) + n *(1 - x) - 6*n *(1 - x) + 11*n*(1 - x) - 6*(1 - x) + n *(1 - x) - 6*n *(1 - x) + 11*n*(1 - x) /
$$\int \frac{x^{2}}{\left(1 - x\right)^{n}}\, dx = C + \begin{cases} - \frac{x^{2}}{2} - x - \log{\left(x - 1 \right)} & \text{for}\: n = 1 \\\frac{x^{2}}{x - 1} + \frac{2 x \log{\left(x - 1 \right)}}{x - 1} - \frac{2 \log{\left(x - 1 \right)}}{x - 1} - \frac{2}{x - 1} & \text{for}\: n = 2 \\- \frac{2 x^{2} \log{\left(x - 1 \right)}}{2 x^{2} - 4 x + 2} + \frac{4 x \log{\left(x - 1 \right)}}{2 x^{2} - 4 x + 2} + \frac{4 x}{2 x^{2} - 4 x + 2} - \frac{2 \log{\left(x - 1 \right)}}{2 x^{2} - 4 x + 2} - \frac{3}{2 x^{2} - 4 x + 2} & \text{for}\: n = 3 \\- \frac{n^{2} x^{3}}{n^{3} \left(1 - x\right)^{n} - 6 n^{2} \left(1 - x\right)^{n} + 11 n \left(1 - x\right)^{n} - 6 \left(1 - x\right)^{n}} + \frac{n^{2} x^{2}}{n^{3} \left(1 - x\right)^{n} - 6 n^{2} \left(1 - x\right)^{n} + 11 n \left(1 - x\right)^{n} - 6 \left(1 - x\right)^{n}} + \frac{3 n x^{3}}{n^{3} \left(1 - x\right)^{n} - 6 n^{2} \left(1 - x\right)^{n} + 11 n \left(1 - x\right)^{n} - 6 \left(1 - x\right)^{n}} - \frac{n x^{2}}{n^{3} \left(1 - x\right)^{n} - 6 n^{2} \left(1 - x\right)^{n} + 11 n \left(1 - x\right)^{n} - 6 \left(1 - x\right)^{n}} - \frac{2 n x}{n^{3} \left(1 - x\right)^{n} - 6 n^{2} \left(1 - x\right)^{n} + 11 n \left(1 - x\right)^{n} - 6 \left(1 - x\right)^{n}} - \frac{2 x^{3}}{n^{3} \left(1 - x\right)^{n} - 6 n^{2} \left(1 - x\right)^{n} + 11 n \left(1 - x\right)^{n} - 6 \left(1 - x\right)^{n}} + \frac{2}{n^{3} \left(1 - x\right)^{n} - 6 n^{2} \left(1 - x\right)^{n} + 11 n \left(1 - x\right)^{n} - 6 \left(1 - x\right)^{n}} & \text{otherwise} \end{cases}$$
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.