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Câu hỏi: $\int\limits_{16}^{55}{\dfrac{dx}{x\sqrt{x+9}}}=a\ln 2+b\ln 5+c\ln 11$ với a, b, c là các số hữu tỉ. Giá trị của $a+b+c$ là:
A. $\dfrac{2}{3}.$
B. $\dfrac{1}{3}.$
C. 1.
D. 2.
A. $\dfrac{2}{3}.$
B. $\dfrac{1}{3}.$
C. 1.
D. 2.
Đặt $t=\sqrt{x+9}\Rightarrow {{t}^{2}}=x+9\Rightarrow 2tdt=dx$.
Đổi cận:
$\begin{aligned}
& \int\limits_{16}^{55}{\dfrac{dx}{x\sqrt{x+9}}}=\int\limits_{5}^{8}{\dfrac{2tdt}{\left( {{t}^{2}}-9 \right)t}}=2\int\limits_{5}^{8}{\dfrac{dt}{{{t}^{2}}-9}}=\dfrac{1}{3}\left( \int\limits_{5}^{8}{\dfrac{dt}{t-3}}-\int\limits_{5}^{8}{\dfrac{dt}{t+3}} \right) \\
& =\dfrac{1}{3}\left( \ln \left| t-3 \right|-\ln \left| t+3 \right| \right)|_{5}^{8}=\dfrac{2}{3}\ln 2+\dfrac{1}{3}\ln 5-\dfrac{1}{3}\ln 11 \\
\end{aligned}$
Vậy $a=\dfrac{2}{3},b=\dfrac{1}{3},c=-\dfrac{1}{3}$.
Đổi cận:
& \int\limits_{16}^{55}{\dfrac{dx}{x\sqrt{x+9}}}=\int\limits_{5}^{8}{\dfrac{2tdt}{\left( {{t}^{2}}-9 \right)t}}=2\int\limits_{5}^{8}{\dfrac{dt}{{{t}^{2}}-9}}=\dfrac{1}{3}\left( \int\limits_{5}^{8}{\dfrac{dt}{t-3}}-\int\limits_{5}^{8}{\dfrac{dt}{t+3}} \right) \\
& =\dfrac{1}{3}\left( \ln \left| t-3 \right|-\ln \left| t+3 \right| \right)|_{5}^{8}=\dfrac{2}{3}\ln 2+\dfrac{1}{3}\ln 5-\dfrac{1}{3}\ln 11 \\
\end{aligned}$
Vậy $a=\dfrac{2}{3},b=\dfrac{1}{3},c=-\dfrac{1}{3}$.
Đáp án A.