Tính tổng T=C202003−C202014+C202025−X202036+...−C202020192022+C202020202023.

Xét khai triển:

x21−x2020=x2∑k=02020C2020k−xk

=x2C20200−C20201x+C20202x2−C20203x3+...−C20202019x2019+C20202020x2020

=C20200x2−C20201x3+C20202x4−C20203x5+...−C20202019x2021+C20202020x2022

Lấy tích phân hai vế ta có:

∫01x21−x2020dx=∫01C20200x2−C20201x3+C20202x4−C20203x5+...−C20202019x2021+C20202020x2022dx

=C20200x33−C20201x44+C20202x55−C20203x66+...−C20202019x20222022+C20202020x2023202310

=13C20200−14C20201+15C20202−16C20203+...−12022C20202019+12023C20202020

Suy ra T=∫01x21−x2020dx.

Đặt t=1−x⇒dt=−dx. Đổi cận x=0⇒t=1x=1⇒t=0. Khi đó ta có:

T=∫01x21−x2020dx=−∫101−t2t2020dt

=∫01t2020t2−2t+1dt=∫01t2022−2t2021+t2020dt

=t20232023−2t20222022+t2021202110

=12023−22022+12021=14133456313

Chọn C.