bai tap 953301122

[953301122] Cho hàm số $f(x)$ liên tục trên $\mathbb{R},$ thỏa mãn $f(x)=f(2020-x)$ và ${\displaystyle \underset{4}{\overset{2016}{\int }}f(x)dx=2.}$ Tính$\underset{4}{\overset{2016}{\int }}xf(x)\text{d}x.$

Xét ${\displaystyle I=\underset{4}{\overset{2016}{\int }}xf(x)dx.}$ Đặt $t=2020-x\Rightarrow dt=-dx$ và $\{\begin{array}{rl}& x=4\Rightarrow t=2016\\ & x=2016\Rightarrow t=4\end{array}$

Do đó ${\displaystyle I=\underset{4}{\overset{2016}{\int }}xf(x)dx=\underset{2016}{\overset{4}{\int }}(2020-t)f(2020-t)(-dt)}$

${\displaystyle =\underset{4}{\overset{2016}{\int }}(2020-x)f(2020-x)dx}$

${\displaystyle =\underset{4}{\overset{2016}{\int }}(2020-x)f(x)dx=2020\underset{4}{\overset{2016}{\int }}f(x)dx-\underset{4}{\overset{2016}{\int }}xf(x)dx=2020.2-I}$

${\displaystyle \Rightarrow I=4040-I\iff 2I=4040\iff I=2020.}$