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Câu hỏi: Cho hàm số $f\left( x \right)$ có đạo hàm liên tục trên $\left[ 0;\pi \right]$. Biết $f\left( 0 \right)=2e$ và $f\left( x \right)$ luôn thỏa mãn đẳng thức ${f}'\left( x \right)+\sin xf\left( x \right)=\cos x{{e}^{\cos x}} \forall x\in \left[ 0;\pi \right]$. Tính $I=\int\limits_{0}^{\pi }{f\left( x \right)dx}$ (làm tròn đến phần trăm)
A. $I\approx 6,55$
B. $I\approx 17,30$
C. $I\approx 10,31$
D. $I\approx 16,91$
A. $I\approx 6,55$
B. $I\approx 17,30$
C. $I\approx 10,31$
D. $I\approx 16,91$
${f}'\left( x \right)+\sin xf\left( x \right)=\cos x{{e}^{\cos x}} \forall x\in \left[ 0;\pi \right]$
$\Leftrightarrow {f}'\left( x \right){{e}^{-\cos x}}+\sin xf\left( x \right){{e}^{-\cos x}}=\cos x\Leftrightarrow {{\left[ f\left( x \right){{e}^{-\cos x}} \right]}^{\prime }}=\cos x$
$\Leftrightarrow \int\limits_{0}^{x}{\left[ f\left( x \right){{e}^{-\cos x}} \right]dx}=\int\limits_{0}^{x}{\cos xdx}\Leftrightarrow \left. f\left( x \right){{e}^{-\cos x}} \right|_{0}^{x}=\left. \sin x \right|_{0}^{x}$
$\Leftrightarrow f\left( x \right){{e}^{-\cos x}}-f\left( 0 \right).{{e}^{-1}}=\sin x\Leftrightarrow f\left( x \right){{e}^{-\cos x}}-2e.{{e}^{-1}}=\sin x$
$\Leftrightarrow f\left( x \right){{e}^{-\cos x}}=\sin x+2\Leftrightarrow f\left( x \right)=\left( \sin x+2 \right){{e}^{\cos x}}$
Khi đó ta có $I=\int\limits_{0}^{\pi }{f\left( x \right)dx}=\int\limits_{0}^{\pi }{\left( \sin x+2 \right){{e}^{\cos x}}dx}\approx 10,31$
$\Leftrightarrow {f}'\left( x \right){{e}^{-\cos x}}+\sin xf\left( x \right){{e}^{-\cos x}}=\cos x\Leftrightarrow {{\left[ f\left( x \right){{e}^{-\cos x}} \right]}^{\prime }}=\cos x$
$\Leftrightarrow \int\limits_{0}^{x}{\left[ f\left( x \right){{e}^{-\cos x}} \right]dx}=\int\limits_{0}^{x}{\cos xdx}\Leftrightarrow \left. f\left( x \right){{e}^{-\cos x}} \right|_{0}^{x}=\left. \sin x \right|_{0}^{x}$
$\Leftrightarrow f\left( x \right){{e}^{-\cos x}}-f\left( 0 \right).{{e}^{-1}}=\sin x\Leftrightarrow f\left( x \right){{e}^{-\cos x}}-2e.{{e}^{-1}}=\sin x$
$\Leftrightarrow f\left( x \right){{e}^{-\cos x}}=\sin x+2\Leftrightarrow f\left( x \right)=\left( \sin x+2 \right){{e}^{\cos x}}$
Khi đó ta có $I=\int\limits_{0}^{\pi }{f\left( x \right)dx}=\int\limits_{0}^{\pi }{\left( \sin x+2 \right){{e}^{\cos x}}dx}\approx 10,31$
Đáp án C.