Google
Câu hỏi: Cho hàm số $f\left( x \right)=\left\{ \begin{aligned}
& 8x-2 \\
& -3{{x}^{3}}+4x+5 \\
\end{aligned} \right.\begin{matrix}
,x>1 \\
,x\le 1 \\
\end{matrix} $. Giả sử $ F\left( x \right) $ là nguyên hàm của $ f\left( x \right) $ trên $ \mathbb{R} $ thỏa $ F\left( 0 \right)=2 $. Giá trị của $ F\left( -1 \right)-F\left( 4 \right)$ bằng:
A. $-64$.
B. $62$.
C. $64$.
D. $-62$
& 8x-2 \\
& -3{{x}^{3}}+4x+5 \\
\end{aligned} \right.\begin{matrix}
,x>1 \\
,x\le 1 \\
\end{matrix} $. Giả sử $ F\left( x \right) $ là nguyên hàm của $ f\left( x \right) $ trên $ \mathbb{R} $ thỏa $ F\left( 0 \right)=2 $. Giá trị của $ F\left( -1 \right)-F\left( 4 \right)$ bằng:
A. $-64$.
B. $62$.
C. $64$.
D. $-62$
Ta có: $F\left( -1 \right)-F\left( 4 \right)=F\left( 0 \right)-\int\limits_{-1}^{0}{f\left( x \right)\text{d}x}-\left( \int\limits_{0}^{4}{f\left( x \right)\text{d}x}+F\left( 0 \right) \right)=-\int\limits_{-1}^{4}{f\left( x \right)\text{d}x}$.
$=-\int\limits_{-1}^{1}{f\left( x \right)\text{d}x}-\int\limits_{1}^{4}{f\left( x \right)\text{d}x}=-\int\limits_{-1}^{1}{\left( -3{{x}^{3}}+4x+5 \right)\text{d}x}-\int\limits_{1}^{4}{\left( 8x-2 \right)\text{d}x}=-64$.
Cách 2:
$F\left( -1 \right)-F\left( 4 \right)=\left[ F\left( -1 \right)-F\left( 1 \right) \right]-\left[ F\left( 4 \right)-F\left( 1 \right) \right]=-\int\limits_{-1}^{1}{f\left( x \right)\text{d}x}-\int\limits_{1}^{4}{f\left( x \right)\text{d}x}$
$=-\int\limits_{-1}^{1}{\left( -3{{x}^{3}}+4x+5 \right)\text{d}x}-\int\limits_{1}^{4}{\left( 8x-2 \right)\text{d}x}=-64$
$=-\int\limits_{-1}^{1}{f\left( x \right)\text{d}x}-\int\limits_{1}^{4}{f\left( x \right)\text{d}x}=-\int\limits_{-1}^{1}{\left( -3{{x}^{3}}+4x+5 \right)\text{d}x}-\int\limits_{1}^{4}{\left( 8x-2 \right)\text{d}x}=-64$.
Cách 2:
$F\left( -1 \right)-F\left( 4 \right)=\left[ F\left( -1 \right)-F\left( 1 \right) \right]-\left[ F\left( 4 \right)-F\left( 1 \right) \right]=-\int\limits_{-1}^{1}{f\left( x \right)\text{d}x}-\int\limits_{1}^{4}{f\left( x \right)\text{d}x}$
$=-\int\limits_{-1}^{1}{\left( -3{{x}^{3}}+4x+5 \right)\text{d}x}-\int\limits_{1}^{4}{\left( 8x-2 \right)\text{d}x}=-64$
Đáp án A.