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Câu hỏi: Cho $\int\limits_{16}^{55}{\dfrac{\text{d}x}{x\sqrt{x+9}}}=a\ln 2+b\ln 5+c\ln 11$ với $a,b,c$ là các số hữu tỉ. Mệnh đề nào dưới đây đúng?
A. $a-b=-c$.
B. $a+b=c$.
C. $a+b=3c$.
D. $a-b=-3c$.
A. $a-b=-c$.
B. $a+b=c$.
C. $a+b=3c$.
D. $a-b=-3c$.
Đặt $t=\sqrt{x+9}$ $\Rightarrow {{t}^{2}}=x+9\Rightarrow 2t\text{d}t=\text{d}x$. Đổi cận: $\left\{ \begin{aligned}
& x=16\to t=5 \\
& x=55\to t=8 \\
\end{aligned} \right.$
Khi đó: $\int\limits_{16}^{55}{\dfrac{\text{d}x}{x\sqrt{x+9}}}$ $=\int\limits_{5}^{8}{\dfrac{2t\text{d}t}{\left( {{t}^{2}}-9 \right)t}}=2\int\limits_{5}^{8}{\dfrac{\text{d}t}{{{t}^{2}}-9}}=\dfrac{1}{3}\left( \int\limits_{5}^{8}{\dfrac{\text{d}t}{t-3}}-\int\limits_{5}^{8}{\dfrac{\text{d}t}{t+3}} \right)$
$=\left. \dfrac{1}{3}\left( \ln \left| x-3 \right|-\ln \left| x+3 \right| \right) \right|_{5}^{8}$ = $\dfrac{2}{3}\ln 2+\dfrac{1}{3}\ln 5-\dfrac{1}{3}\ln 11$.
Vậy $a=\dfrac{2}{3}$, $b=\dfrac{1}{3}$, $c=-\dfrac{1}{3}$. Mệnh đề $a-b=-c$ đúng.
& x=16\to t=5 \\
& x=55\to t=8 \\
\end{aligned} \right.$
Khi đó: $\int\limits_{16}^{55}{\dfrac{\text{d}x}{x\sqrt{x+9}}}$ $=\int\limits_{5}^{8}{\dfrac{2t\text{d}t}{\left( {{t}^{2}}-9 \right)t}}=2\int\limits_{5}^{8}{\dfrac{\text{d}t}{{{t}^{2}}-9}}=\dfrac{1}{3}\left( \int\limits_{5}^{8}{\dfrac{\text{d}t}{t-3}}-\int\limits_{5}^{8}{\dfrac{\text{d}t}{t+3}} \right)$
$=\left. \dfrac{1}{3}\left( \ln \left| x-3 \right|-\ln \left| x+3 \right| \right) \right|_{5}^{8}$ = $\dfrac{2}{3}\ln 2+\dfrac{1}{3}\ln 5-\dfrac{1}{3}\ln 11$.
Vậy $a=\dfrac{2}{3}$, $b=\dfrac{1}{3}$, $c=-\dfrac{1}{3}$. Mệnh đề $a-b=-c$ đúng.
Đáp án A.