Интеграл x*((1-x)^n) (dx)

Вы ввели

  1              
  /              
 |               
 |           n   
 |  x*(1 - x)  dx
 |               
/                
0                

$$\int_{0}^{1} x \left(- x + 1\right)^{n}\, dx$$

Ответ

  1                   1              
  /                   /              
 |                   |               
 |           n       |           n   
 |  x*(1 - x)  dx =  |  x*(1 - x)  dx
 |                   |               
/                   /                
0                   0                

$$\int_{0}^{1} x \left(- x + 1\right)^{n}\, dx = \int_{0}^{1} x \left(- x + 1\right)^{n}\, dx$$

Ответ (Неопределённый)

                       //               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            /

$${{\left(\left(n+1\right)\,x^2-n\,x-1\right)\,e^{n\,\log \left(1-x \right)}}\over{n^2+3\,n+2}}$$