Đáp án:
$I = - \dfrac{2^{2023}}{4090506}$
Giải thích các bước giải:
$\quad I = \displaystyle\int\limits_0^2x(x-2)^{2021}dx$
Đặt $u = x - 2 \Rightarrow u + 2 = x$
$\Rightarrow du = dx$
Đổi cận:
$\begin{array}{c|ccc}x&0&&&2\\\hline u&-2&&&0\\\end{array}$
Ta được:
$\quad I = \displaystyle\int\limits_{-2}^0(u+2)u^{2021}du$
$\Leftrightarrow I = \displaystyle\int\limits_{-2}^0\left(u^{2022} + 2u^{2021}\right)du$
$\Leftrightarrow I = \dfrac{u^{2023}}{2023}\Bigg|_{-2}^0 +2\cdot \dfrac{u^{2022}}{2022}\Bigg|_{-2}^0$
$\Leftrightarrow I = \dfrac{2^{2023}}{2023} - 2\cdot \dfrac{2^{2022}}{2022}$
$\Leftrightarrow I = - \dfrac{2^{2023}}{4090506}$
