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Câu hỏi: Cho hàm số f(x) liên tục trên đoạn $\left[ 0;\dfrac{\pi }{3} \right]$. Biết ${f}'\left( x \right).\cos x+f\left( x \right).\sin x=1$ với $\forall x\in \left[ 0;\dfrac{\pi }{3} \right]$ và $f\left( 0 \right)=1.$ Tính $I=\int\limits_{0}^{\dfrac{\pi }{3}}{f\left( x \right)dx}.$
A. $I=\dfrac{\sqrt{3}+1}{2}$
B. $I=\dfrac{\sqrt{3}-1}{2}$
C. $I=\dfrac{1}{2}$
D. $I=\dfrac{1}{2}+\dfrac{\pi }{3}$
A. $I=\dfrac{\sqrt{3}+1}{2}$
B. $I=\dfrac{\sqrt{3}-1}{2}$
C. $I=\dfrac{1}{2}$
D. $I=\dfrac{1}{2}+\dfrac{\pi }{3}$
Ta có ${{\left[ \dfrac{f\left( x \right)}{\cos x} \right]}^{'}}=\dfrac{f'\left( x \right).\cos x+f\left( x \right).\sin x}{{{\cos }^{2}}x}=\dfrac{1}{{{\cos }^{2}}x}$.
$\Rightarrow \dfrac{f\left( x \right)}{\cos x}=\int{\dfrac{1}{{{\cos }^{2}}x}dx}=\tan x+C$.
Mà $f\left( 0 \right)=1\Rightarrow C=1\Rightarrow f\left( x \right)=\cos x\left( \tan x+1 \right)=\sin x+\cos x$
$\Rightarrow I=\int\limits_{0}^{\dfrac{\pi }{3}}{\left( \sin x+\cos x \right)dx}=\left( -\cos x+\sin x \right)\left| \begin{aligned}
& ^{\dfrac{\pi }{3}} \\
& _{0} \\
\end{aligned} \right.=\dfrac{1+\sqrt{3}}{2}$.
$\Rightarrow \dfrac{f\left( x \right)}{\cos x}=\int{\dfrac{1}{{{\cos }^{2}}x}dx}=\tan x+C$.
Mà $f\left( 0 \right)=1\Rightarrow C=1\Rightarrow f\left( x \right)=\cos x\left( \tan x+1 \right)=\sin x+\cos x$
$\Rightarrow I=\int\limits_{0}^{\dfrac{\pi }{3}}{\left( \sin x+\cos x \right)dx}=\left( -\cos x+\sin x \right)\left| \begin{aligned}
& ^{\dfrac{\pi }{3}} \\
& _{0} \\
\end{aligned} \right.=\dfrac{1+\sqrt{3}}{2}$.
Đáp án A.