Tích phân $\Large \int_{-2}^{2} \dfrac{x^{2020}}{e^{x}+1} d x=\dfrac{2

Tích phân $\Large \int_{-2}^{2} \dfrac{x^{2020}}{e^{x}+1} d x=\dfrac{2^{a}}{b}$. Tính tổng $\Large S=a+b$

• A. $\Large S=0$
• B. $\Large S=2021$
• C. $\Large S=2020$
• D. $\Large S=4042$

Chọn D

Xét $\Large I=\int_{-2}^{2} \dfrac{x^{2020}}{e^{x}+1} d x$

Đặt $\Large x=-t \Rightarrow d x=- d t$

Đổi cận $\Large x=-2 \Rightarrow t=2 ; x=2 \Rightarrow t=-2$

Ta được $\Large I=\int_{2}^{-2} \dfrac{(-t)^{2020}}{e^{-t}+1} t(- d t)=\int_{-2}^{2} \dfrac{t^{2020}}{\dfrac{1}{e^{t}}+1} d t=\int_{-2}^{2} \dfrac{t^{2020}  e^{t}}{e^{t}+1} \c d t=\int_{-2}^{2} \dfrac{x^{2020}  e^{x}}{e^{x}+1}  d x$

Suy ra $\Large 2 I=I+I=\int_{-2}^{2} \dfrac{x^{2020}}{e^{x}+1} d x+\int_{-2}^{2} \dfrac{x^{2020}  e^{x}}{e^{x}+1}  d x$ $\Large =\int_{-2}^{2} x^{2020}  d x=\left.\dfrac{x^{2021}}{2021}\right|_{-2} ^{2}=\dfrac{2^{2021}-(-2)^{2021}}{2021}=\dfrac{2^{2022}}{2021}$

Do đó $\Large I=\dfrac{2^{2021}}{2021}$. Suy ra $\Large a=b=2021$, Vậy $\Large S=a+b=4042$