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Câu hỏi: Cho hàm số $f$ liên tục, $f(x)>-1, f(0)=0$ và thỏa mãn $f^{\prime}(x) \sqrt{x^2+1}=2 x \sqrt{f(x)+1}$. Tính $f(\sqrt{3})$.
A. 0 .
B. 3 .
C. 7 .
D. 9 .
A. 0 .
B. 3 .
C. 7 .
D. 9 .
Ta có $f^{\prime}(x) \sqrt{x^2+1}=2 x \sqrt{f(x)+1} \Leftrightarrow \dfrac{f^{\prime}(x)}{\sqrt{f(x)+1}}=\dfrac{2 x}{\sqrt{x^2+1}}$
$
\begin{aligned}
& \Leftrightarrow \int_0^{\sqrt{3}} \dfrac{f^{\prime}(x)}{\sqrt{f(x)+1}} \mathrm{~d} x=\left.\int_0^{\sqrt{3}} \dfrac{2 x}{\sqrt{x^2+1}} \mathrm{~d} x \Leftrightarrow \sqrt{f(x)+1}\right|_0 ^{\sqrt{3}}=\left.\left.\sqrt{x^2+1}\right|_0 ^{\sqrt{3}} \Leftrightarrow \sqrt{f(x)+1}\right|_0 ^{\sqrt{3}}=1 \\
\Leftrightarrow & \sqrt{f(\sqrt{3})+1}-\sqrt{f(0)+1}=1 \Leftrightarrow \sqrt{f(\sqrt{3})+1}=2 \Leftrightarrow f(\sqrt{3})=3 .
\end{aligned}
$
$
\begin{aligned}
& \Leftrightarrow \int_0^{\sqrt{3}} \dfrac{f^{\prime}(x)}{\sqrt{f(x)+1}} \mathrm{~d} x=\left.\int_0^{\sqrt{3}} \dfrac{2 x}{\sqrt{x^2+1}} \mathrm{~d} x \Leftrightarrow \sqrt{f(x)+1}\right|_0 ^{\sqrt{3}}=\left.\left.\sqrt{x^2+1}\right|_0 ^{\sqrt{3}} \Leftrightarrow \sqrt{f(x)+1}\right|_0 ^{\sqrt{3}}=1 \\
\Leftrightarrow & \sqrt{f(\sqrt{3})+1}-\sqrt{f(0)+1}=1 \Leftrightarrow \sqrt{f(\sqrt{3})+1}=2 \Leftrightarrow f(\sqrt{3})=3 .
\end{aligned}
$
Đáp án B.