$I=\displaystyle\int_{16}^{55}\dfrac{dx}{x\sqrt{x+9}}$
Đặt $t=\sqrt{x+9}$
$\to t^2=x+9$
$\to 2tdt=dx$
Đổi cận:
$x=16\to t=5$
$x=55\to t=8$
$I=\displaystyle\int_{5}^{8}\dfrac{1}{t(t^2-9)}.2tdt$
$I=\displaystyle\int_{5}^{8}\dfrac{2}{(t-3)(t+3)}dt$
$=2\displaystyle\int_{5}^{8}\dfrac{t+3-(t-3)}{(t-3)(t+3)}.\dfrac{1}{6}dt$
$=\dfrac{1}{3}\displaystyle\int_{5}^{8}\left( \dfrac{1}{t-3}-\dfrac{1}{t+3}\right)dt$
$=\dfrac{1}{3}\ln\left| \dfrac{t-3}{t+3}\right| |_{5}^{8}$
$=\dfrac{1}{3}(\ln\dfrac{5}{11}-\ln 4^{-1})$
$=\dfrac{1}{3}(\ln5-\ln11+\ln4)$
$=\dfrac{1}{3}\ln5-\dfrac{1}{3}\ln11+\dfrac{2}{3}\ln2$
$\to \begin{cases} a=\dfrac{2}{3}\\ b=\dfrac{1}{3}\\c=\dfrac{-1}{3}\end{cases}$
