Đáp án:
$M = \ln\left|\sqrt{x+1} -1\right| - \ln\left|\sqrt{x+1} +1\right| + C$
Giải thích các bước giải:
$\quad M =\displaystyle\int\dfrac{dx}{x\sqrt{x+1}}$
Đặt $u = \sqrt{x +1}$
$\to du = \dfrac{dx}{2\sqrt{x+1}}$
Ta được:
$M = 2\displaystyle\int\dfrac{du}{u^2 -1}$
$\to M =2\displaystyle\int\dfrac{du}{(u-1)(u+1)}$
$\to M = \displaystyle\int\left(\dfrac{1}{u-1} -\dfrac{1}{u+1}\right)du$
$\to M = \displaystyle\int\dfrac{du}{u-1} - \displaystyle\int\dfrac{du}{u+1}$
$\to M = \displaystyle\int\dfrac{d(u-1)}{u-1} - \displaystyle\int\dfrac{d(u+1)}{u+1}$
$\to M =\ln|u-1| -\ln|u+1| + C$
$\to M = \ln\left|\sqrt{x+1} -1\right| - \ln\left|\sqrt{x+1} +1\right| + C$
