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Câu hỏi: Cho hàm số $f\left( x \right)$ có $f\left( 3 \right)=3$ và ${f}'\left( x \right)=\dfrac{x}{x+1-\sqrt{x+1}}$, $\forall x>0$. Khi đó $\int\limits_{3}^{8}{f\left( x \right)\text{d}x}$ bằng
A. $7$.
B. $\dfrac{197}{6}$.
C. $\dfrac{29}{2}$.
D. $\dfrac{181}{6}$.
A. $7$.
B. $\dfrac{197}{6}$.
C. $\dfrac{29}{2}$.
D. $\dfrac{181}{6}$.
Xét $\int{{f}'\left( x \right)\text{d}x}=\int{\dfrac{x}{x+1-\sqrt{x+1}}\text{d}x}$. Đặt $t=\sqrt{x+1}\Rightarrow x+1={{t}^{2}}\Rightarrow x={{t}^{2}}-1\Rightarrow \text{d}x=2t\text{d}t$.
Khi đó, $\int{{f}'\left( x \right)\text{d}x}=\int{\dfrac{x}{x+1-\sqrt{x+1}}\text{d}x}=\int{\dfrac{{{t}^{2}}-1}{{{t}^{2}}-t}\cdot 2t\text{d}t}=\int{\dfrac{\left( t-1 \right).\left( t+1 \right)}{t.\left( t-1 \right)}\cdot 2t\text{d}t}=\int{\left( 2t+2 \right)\text{d}t}$
$={{t}^{2}}+2t+C=\left( x+1 \right)+2\sqrt{x+1}+C$.
Mà $f\left( 3 \right)=3\Leftrightarrow \left( 3+1 \right)+2\sqrt{3+1}+C=3\Leftrightarrow C=-5$.
$\Rightarrow f\left( x \right)=\left( x+1 \right)+2\sqrt{x+1}-5=x+2\sqrt{x+1}-4$.
$\Rightarrow \int\limits_{3}^{8}{f\left( x \right)\text{d}x}=\int\limits_{3}^{8}{\left( x+2\sqrt{x+1}-4 \right)\text{d}x}=\left. \left( \dfrac{{{x}^{2}}}{2}+\dfrac{4}{3}\sqrt{{{\left( x+1 \right)}^{3}}}-4x \right) \right|_{3}^{8}=36-\dfrac{19}{6}=\dfrac{197}{6}$.
Khi đó, $\int{{f}'\left( x \right)\text{d}x}=\int{\dfrac{x}{x+1-\sqrt{x+1}}\text{d}x}=\int{\dfrac{{{t}^{2}}-1}{{{t}^{2}}-t}\cdot 2t\text{d}t}=\int{\dfrac{\left( t-1 \right).\left( t+1 \right)}{t.\left( t-1 \right)}\cdot 2t\text{d}t}=\int{\left( 2t+2 \right)\text{d}t}$
$={{t}^{2}}+2t+C=\left( x+1 \right)+2\sqrt{x+1}+C$.
Mà $f\left( 3 \right)=3\Leftrightarrow \left( 3+1 \right)+2\sqrt{3+1}+C=3\Leftrightarrow C=-5$.
$\Rightarrow f\left( x \right)=\left( x+1 \right)+2\sqrt{x+1}-5=x+2\sqrt{x+1}-4$.
$\Rightarrow \int\limits_{3}^{8}{f\left( x \right)\text{d}x}=\int\limits_{3}^{8}{\left( x+2\sqrt{x+1}-4 \right)\text{d}x}=\left. \left( \dfrac{{{x}^{2}}}{2}+\dfrac{4}{3}\sqrt{{{\left( x+1 \right)}^{3}}}-4x \right) \right|_{3}^{8}=36-\dfrac{19}{6}=\dfrac{197}{6}$.
Đáp án B.