Đáp án:
\(I = \dfrac{3^{2021} - 2^{2021}}{4042}\)
Giải thích các bước giải:
\(\begin{array}{l}
\quad I = \displaystyle\int\limits_2^3\dfrac{(x+1)^{2020}}{(x-1)^{2022}}dx\\
\to I = - \dfrac12\displaystyle\int\limits_2^3\left(\dfrac{x+1}{x-1}\right)^{2020}\cdot \dfrac{-2}{(x-1)^{2}}dx\\
\text{Đặt}\ t = \dfrac{x+1}{x-1}\\
\Rightarrow dt = \dfrac{-2}{(x-1)^2}dx\\
\text{Đổi cận:}\\
x\quad \Big|\quad 2\qquad 3\\
\overline{\ t\quad\Big|\quad 3\qquad 2}\\
\text{Ta được:}\\
\quad I = - \dfrac12\displaystyle\int\limits_3^2t^{2020}dt\\
\to I = \dfrac12\cdot \dfrac{t^{2021}}{2021}\Bigg|_2^3\\
\to I = \dfrac{3^{2021} - 2^{2021}}{4042}
\end{array}\)
