Cho hàm số \(y=f\left( x \right)\) liên tục trên \(\mathbb{R}\) thỏa mãn \(xf\left( {{x}^{2}} \right)-f\left( 2x \right)=2{{x}^{3}}+2x,\,\,\,\forall x\in \mathbb{R}\). Tính giá trị \(I=\int\limits_{1}^{2}{f\left( x \right)\text{d}x}\).

A. I = 25

B. I = 21

C. I = 27

D. I = 23

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Chủ đề: Đề thi THPT QG

Môn: Toán

Lời giải:

Báo sai

\(xf\left( {{x}^{2}} \right)-f\left( 2x \right)=2{{x}^{3}}+2x\Rightarrow \int\limits_{1}^{2}{\left[ xf\left( {{x}^{2}} \right)-f\left( 2x \right) \right]}\text{d}x=\int\limits_{1}^{2}{\left( 2{{x}^{3}}+2x \right)\text{d}x}\)

\(\Leftrightarrow \int\limits_{1}^{2}{\left[ xf\left( {{x}^{2}} \right) \right]}\text{d}x-\int\limits_{1}^{2}{\left[ f\left( 2x \right) \right]}\text{d}x=\left( \frac{{{x}^{4}}}{2}+{{x}^{2}} \right)\left| \begin{align} & 2 \\ & 1 \\ \end{align} \right.\Leftrightarrow \int\limits_{1}^{2}{\left[ xf\left( {{x}^{2}} \right) \right]}\text{d}x-\int\limits_{1}^{2}{\left[ f\left( 2x \right) \right]}\text{d}x=\frac{21}{2}\).

+ Tính \(\int\limits_{1}^{2}{\left[ xf\left( {{x}^{2}} \right) \right]}\text{d}x\)

Đặt \(u={{x}^{2}}\Rightarrow \text{d}u=2x\text{d}x\Leftrightarrow x\text{d}x=\frac{\text{d}u}{2}\)

\(x=1\Rightarrow u=1;\,\,x=2\Rightarrow u=4\)

Suy ra \(\int\limits_{1}^{2}{\left[ xf\left( {{x}^{2}} \right) \right]}\text{d}x=\int\limits_{1}^{4}{\frac{f\left( u \right)}{2}\text{d}u}=\frac{1}{2}\int\limits_{1}^{4}{f\left( x \right)\text{d}x}\)

+ Tính \(\int\limits_{1}^{2}{\left[ f\left( 2x \right) \right]}\text{d}x\)

Đặt \(t=2x\Rightarrow \text{d}t=2\text{d}x\Leftrightarrow \text{d}x=\frac{\text{d}t}{2}\).

\(x=1\Rightarrow t=2;\,\,x=2\Rightarrow t=4\).

Suy ra \(\int\limits_{1}^{2}{\left[ f\left( 2x \right) \right]}\text{d}x=\int\limits_{2}^{4}{\frac{f\left( t \right)}{2}\text{d}t}=\frac{1}{2}\int\limits_{2}^{4}{f\left( x \right)\text{d}x}\)

Thay vào ta được \(\frac{1}{2}\int\limits_{1}^{4}{f\left( x \right)\text{d}x}-\frac{1}{2}\int\limits_{2}^{4}{f\left( x \right)\text{d}x}=\frac{21}{2}\Leftrightarrow \frac{1}{2}\int\limits_{1}^{2}{f\left( x \right)\text{d}x}+\frac{1}{2}\int\limits_{2}^{4}{f\left( x \right)\text{d}x}-\frac{1}{2}\int\limits_{2}^{4}{f\left( x \right)\text{d}x=\frac{21}{2}}\)

\(\Leftrightarrow \frac{1}{2}\int\limits_{1}^{2}{f\left( x \right)\text{d}x}=\frac{21}{2}\Leftrightarrow \int\limits_{1}^{2}{f\left( x \right)\text{d}x}=21\).

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