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Câu hỏi: Cho hàm số $f\left( x \right)$ 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 ${{\left[ \dfrac{f\left( x \right)}{\cos x} \right]}^{\prime }}=\dfrac{{f}'\left( x \right).\cos x+f\left( x \right).\sin x}{{{\cos }^{2}}x}=\dfrac{1}{{{\cos }^{2}}x}$
$\Rightarrow \dfrac{f\left( x \right)}{\cos x}=\int{\dfrac{1}{{{\cos }^{2}}x}dx}=\tan x+C.$ Mà
$f\left( 0 \right)=1\Rightarrow C=1\Rightarrow f\left( x \right)=\cos x\left( \tan x+1 \right)=\sin x+\cos x$
$\Rightarrow I=\int\limits_{0}^{\dfrac{\pi }{3}}{\left( \sin x+\cos x \right)=\left( -\cos x+\sin x \right)\left| _{\begin{smallmatrix}
\\
0
\end{smallmatrix}}^{\dfrac{\pi }{3}} \right.}=\dfrac{1+\sqrt{3}}{2}.$
$\Rightarrow \dfrac{f\left( x \right)}{\cos x}=\int{\dfrac{1}{{{\cos }^{2}}x}dx}=\tan x+C.$ Mà
$f\left( 0 \right)=1\Rightarrow C=1\Rightarrow f\left( x \right)=\cos x\left( \tan x+1 \right)=\sin x+\cos x$
$\Rightarrow I=\int\limits_{0}^{\dfrac{\pi }{3}}{\left( \sin x+\cos x \right)=\left( -\cos x+\sin x \right)\left| _{\begin{smallmatrix}
\\
0
\end{smallmatrix}}^{\dfrac{\pi }{3}} \right.}=\dfrac{1+\sqrt{3}}{2}.$
Đáp án A.