Question

[Lê Thị Huệ]

Tính tích phân từ 0 đến 100 của x(x-1)(x-2)....(x-100)

Answer 1

[Nguyễn Việt Lâm]

\(I=\int\limits^{100}_0x\left(x-1\right)...\left(x-100\right)dx\)

Đặt \(100-x=t\Rightarrow dx=-dt;\left\{{}\begin{matrix}x=0\Rightarrow t=100\\x=100\Rightarrow t=0\end{matrix}\right.\)

\(\Rightarrow I=\int\limits^0_{100}\left(100-t\right)\left(99-t\right)...\left(1-t\right)\left(-t\right).\left(-dt\right)\)

\(I=\int\limits^0_{100}\left(-1\right)\left(t-100\right).\left(-1\right)\left(t-99\right)...\left(-1\right)\left(t-1\right)\left(-1\right)t\left(-dt\right)\) (101 số -1)

\(I=-\int\limits^0_{100}t\left(t-1\right)\left(t-2\right)...\left(t-100\right)\left(-dt\right)\)

\(I=-\int\limits^{100}_0t\left(t-1\right)\left(t-2\right)...\left(t-100\right)dt\)

\(I=-\int\limits^{100}_0x\left(x-1\right)\left(x-2\right)...\left(x-100\right)dx=-I\)

\(\Rightarrow2I=0\Rightarrow I=0\)