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Resolución de integrales/ (1-x)/ x^2÷(1-x)^n

Integral de x^2÷(1-x)^n dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1            
  /            
 |             
 |      2      
 |     x       
 |  -------- dx
 |         n   
 |  (1 - x)    
 |             
/              
0
∫01x2(1−x)n dx\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|
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                     ||                                                                                                                                                                                                        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                                                                                                                                                                                     |
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                     ||                                                                                           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)            /
∫x2(1−x)n dx=C+{−x22−x−log⁡(x−1)for n=1x2x−1+2xlog⁡(x−1)x−1−2log⁡(x−1)x−1−2x−1for n=2−2x2log⁡(x−1)2x2−4x+2+4xlog⁡(x−1)2x2−4x+2+4x2x2−4x+2−2log⁡(x−1)2x2−4x+2−32x2−4x+2for n=3−n2x3n3(1−x)n−6n2(1−x)n+11n(1−x)n−6(1−x)n+n2x2n3(1−x)n−6n2(1−x)n+11n(1−x)n−6(1−x)n+3nx3n3(1−x)n−6n2(1−x)n+11n(1−x)n−6(1−x)n−nx2n3(1−x)n−6n2(1−x)n+11n(1−x)n−6(1−x)n−2nxn3(1−x)n−6n2(1−x)n+11n(1−x)n−6(1−x)n−2x3n3(1−x)n−6n2(1−x)n+11n(1−x)n−6(1−x)n+2n3(1−x)n−6n2(1−x)n+11n(1−x)n−6(1−x)notherwise\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}
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    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

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