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

Integral de x(1-x)^n dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1              
  /              
 |               
 |           n   
 |  x*(1 - x)  dx
 |               
/                
0
∫01x(1−x)n dx\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            /
∫x(1−x)n dx=C+{xlog⁡(x−1)x−1−log⁡(x−1)x−1−1x−1for n=−2−x−log⁡(x−1)for n=−1nx2(1−x)nn2+3n+2−nx(1−x)nn2+3n+2+x2(1−x)nn2+3n+2−(1−x)nn2+3n+2otherwise\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}
Simplificar
Respuesta [src] 
/ oo - pi*I    for n = -2
|                        
| oo + pi*I    for n = -1
|                        
<     1                  
|------------  otherwise 
|     2                  
|2 + n  + 3*n            
\
{∞−iπfor n=−2∞+iπfor n=−11n2+3n+2otherwise\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            
\
{∞−iπfor n=−2∞+iπfor n=−11n2+3n+2otherwise\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.

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