Google
Câu hỏi: Cho $f\left( x \right)$, $g\left( x \right)$ là các hàm số liên tục trên $\mathbb{R}$. Trong các mệnh đề sau, mệnh đề nào sai?
A. $\int{2f\left( x \right)}\text{d}x\text{ = 2}\int{f\left( x \right)\text{d}}x$.
B. $\int{f\left( x \right)g\left( x \right)\text{d}}x=\int{f\left( x \right)\text{d}x.\int{g\left( x \right)\text{d}x}}$.
C. $\int{\left[ f\left( x \right)+g\left( x \right) \right]\text{d}x=\int{f\left( x \right)\text{d}x+\int{g\left( x \right)\text{d}x}}}$.
D. $\int{\left[ f\left( x \right)-g\left( x \right) \right]\text{d}x=\int{f\left( x \right)\text{d}x-\int{g\left( x \right)\text{d}x}}}$.
A. $\int{2f\left( x \right)}\text{d}x\text{ = 2}\int{f\left( x \right)\text{d}}x$.
B. $\int{f\left( x \right)g\left( x \right)\text{d}}x=\int{f\left( x \right)\text{d}x.\int{g\left( x \right)\text{d}x}}$.
C. $\int{\left[ f\left( x \right)+g\left( x \right) \right]\text{d}x=\int{f\left( x \right)\text{d}x+\int{g\left( x \right)\text{d}x}}}$.
D. $\int{\left[ f\left( x \right)-g\left( x \right) \right]\text{d}x=\int{f\left( x \right)\text{d}x-\int{g\left( x \right)\text{d}x}}}$.
Đáp án A, C, D đúng, đó là các tính chất của nguyên hàm, suy ra đáp án B sai.
Phản ví dụ cho đáp án B.
Đặt $h\left( x \right)=$ $\int{x\left( x+1 \right)\text{d}x}$ $=\int{\left( {{x}^{2}}+x \right)\text{d}x}=\frac{{{x}^{3}}}{3}+\frac{{{x}^{2}}}{2}+C$.
$k\left( x \right)=$ $\int{x\text{d}x.\int{\left( x+1 \right)\text{d}x}}$ $=\left( \frac{{{x}^{2}}}{2}+{{C}_{1}} \right)\left( \frac{{{x}^{2}}}{2}+x+{{C}_{2}} \right)$ $\Rightarrow h\left( x \right)\ne k\left( x \right)$
$\Rightarrow \int{x\left( x+1 \right)\text{d}x}\ne \int{x\text{d}x.\int{\left( x+1 \right)\text{d}x}}$.
Phản ví dụ cho đáp án B.
Đặt $h\left( x \right)=$ $\int{x\left( x+1 \right)\text{d}x}$ $=\int{\left( {{x}^{2}}+x \right)\text{d}x}=\frac{{{x}^{3}}}{3}+\frac{{{x}^{2}}}{2}+C$.
$k\left( x \right)=$ $\int{x\text{d}x.\int{\left( x+1 \right)\text{d}x}}$ $=\left( \frac{{{x}^{2}}}{2}+{{C}_{1}} \right)\left( \frac{{{x}^{2}}}{2}+x+{{C}_{2}} \right)$ $\Rightarrow h\left( x \right)\ne k\left( x \right)$
$\Rightarrow \int{x\left( x+1 \right)\text{d}x}\ne \int{x\text{d}x.\int{\left( x+1 \right)\text{d}x}}$.
Đáp án B.