Đề bài: Cho hàm số \(y = f\left( x \right)\) liên tục trên \(\mathbb{R}\) và thỏa mãn \(f\left( x \right) + f\left( { – x} \right) = {x^2},\forall x \in \mathbb{R}\) . Tính \(I = \int\limits_{ – 1}^1 {f\left( x \right)} dx.\)

Ta có \(f\left( x \right) + f\left( { – x} \right) = {x^2} \Rightarrow \int\limits_{ – 1}^1 {\left[ {f\left( x \right) + f\left( { – x} \right)} \right]} dx = \int\limits_{ – 1}^1 {{x^2}dx} \)

\( \Leftrightarrow \int\limits_{ – 1}^1 {f\left( x \right)} dx + \int\limits_{ – 1}^1 {f\left( { – x} \right)} dx = \int\limits_{ – 1}^1 {{x^2}} dx\)

Đặt \(t =  – x \Rightarrow dt =  – dx \Rightarrow \left\{ {\begin{array}{*{20}{c}}{x =  – 1,t = 1}\\{x = 1,t =  – 1}\end{array}} \right. \Rightarrow \int\limits_{ – 1}^1 {f\left( { – x} \right)} dx =  – \int\limits_1^{ – 1} {f\left( t \right)dt}  = \int\limits_{ – 1}^1 {f\left( t \right)dt} \) \( = \int\limits_{ – 1}^1 {f\left( x \right)} dx\)

Suy ra \(\int\limits_{ – 1}^1 {f\left( x \right)} dx + \int\limits_{ – 1}^1 {f\left( x \right)} dx = \int\limits_{ – 1}^1 {{x^2}dx}  \Leftrightarrow 2\int\limits_{ – 1}^1 {f\left( x \right)} dx = \frac{{{x^3}}}{3}\left| {\begin{array}{*{20}{c}}1\\{ – 1}\end{array}} \right. = \frac{2}{3} \Rightarrow \int\limits_{ – 1}^1 {f\left( x \right)} dx = \frac{1}{3}.\)