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Câu hỏi: Cho hàm số $y=f\left( x \right)$ có đạo hàm liên tục trên ${\mathbb{R}}$ thỏa mãn $f\left( 0 \right)=0$ và $f\left( x \right)+f\prime \left( x \right)=\sin x+x\cdot \sin x+x\cdot \cos x, \forall x\in \mathbb{R}$. Diện tích hình phẳng giới hạn bởi các đường $y=f\left( x \right)$, trục hoành, trục tung và ${x=\dfrac{\pi}{2}}$ bằng
A. ${\pi}$.
B. ${\dfrac{\pi}{2}}$.
C. $1$.
D. $2$.
A. ${\pi}$.
B. ${\dfrac{\pi}{2}}$.
C. $1$.
D. $2$.
Ta có $f\left( x \right)+f\prime \left( x \right)=\sin x+x\cdot \sin x+x\cdot \cos x$
$\Leftrightarrow {{e}^{x}}.f\left( x \right)+{{e}^{x}}.f\prime \left( x \right)=\left( \sin x+x\cdot \sin x+x\cdot \cos x \right){{e}^{x}}$
$\Leftrightarrow {{\left[ {{e}^{x}}.f\left( x \right) \right]}^{\prime }}=\left( {{e}^{x}}\sin x+{{e}^{x}}x\cdot \sin x+{{e}^{x}}x\cdot \cos x \right)$
$\Leftrightarrow {{\left[ {{e}^{x}}.f\left( x \right) \right]}^{\prime }}={{\left( {{e}^{x}}.x\cdot \sin x \right)}^{\prime }}$.
Suy ra $\int{{{\left[ {{e}^{x}}.f\left( x \right) \right]}^{\prime }}\text{d}x}=\int{{{\left( {{e}^{x}}x\cdot \sin x \right)}^{\prime }}\text{d}x}\Leftrightarrow {{e}^{x}}.f\left( x \right)={{e}^{x}}x\cdot \sin x+C$.
Do $f\left( 0 \right)=0$ suy ra $C=0$. Khi đó ${{e}^{x}}.f\left( x \right)={{e}^{x}}x\cdot \sin x\Leftrightarrow f\left( x \right)=x\cdot \sin x$.
Suy ra $S=\int\limits_{0}^{\dfrac{\pi }{2}}{\left| f\left( x \right) \right|\text{d}x}=\int\limits_{0}^{\dfrac{\pi }{2}}{\left| x\sin x \right|\text{d}x}=1$.
$\Leftrightarrow {{e}^{x}}.f\left( x \right)+{{e}^{x}}.f\prime \left( x \right)=\left( \sin x+x\cdot \sin x+x\cdot \cos x \right){{e}^{x}}$
$\Leftrightarrow {{\left[ {{e}^{x}}.f\left( x \right) \right]}^{\prime }}=\left( {{e}^{x}}\sin x+{{e}^{x}}x\cdot \sin x+{{e}^{x}}x\cdot \cos x \right)$
$\Leftrightarrow {{\left[ {{e}^{x}}.f\left( x \right) \right]}^{\prime }}={{\left( {{e}^{x}}.x\cdot \sin x \right)}^{\prime }}$.
Suy ra $\int{{{\left[ {{e}^{x}}.f\left( x \right) \right]}^{\prime }}\text{d}x}=\int{{{\left( {{e}^{x}}x\cdot \sin x \right)}^{\prime }}\text{d}x}\Leftrightarrow {{e}^{x}}.f\left( x \right)={{e}^{x}}x\cdot \sin x+C$.
Do $f\left( 0 \right)=0$ suy ra $C=0$. Khi đó ${{e}^{x}}.f\left( x \right)={{e}^{x}}x\cdot \sin x\Leftrightarrow f\left( x \right)=x\cdot \sin x$.
Suy ra $S=\int\limits_{0}^{\dfrac{\pi }{2}}{\left| f\left( x \right) \right|\text{d}x}=\int\limits_{0}^{\dfrac{\pi }{2}}{\left| x\sin x \right|\text{d}x}=1$.
Đáp án C.