=$\int\limits\frac{x(\sqrt{x}-\sqrt{x+1})}{(\sqrt{x}+\sqrt{x+1}).(\sqrt{x}-\sqrt{x+1})}$dx= $-\int\limits x\sqrt{x}$dx + $\int\limits x.\sqrt{x+1}$dx
= $ A + B$
+) A= $-\int\limits x.x^{\frac{1}{2}}$dx = $-\int\limits x^{\frac{3}{2}}$dx= $-\frac{2}{5}.$$x^{\frac{5}{2}}$
B=$\int\limits x.\sqrt{x+1}$dx
đặt t=$\sqrt{x+1}$ $\Rightarrow t^{2}$= $x+1$ $\Rightarrow 2tdt=dx$
+) B= $2\int\limits (t^{2}-1).t.tdt$ $=2\int\limits t^{4} dt$ $-2\int\limits t^{2}dt$ $=\frac{2}{5}.t^{5}-\frac{2}{3}.t^{3}$
=$\frac{2}{5}.\sqrt{(x+1)^{5}}-\frac{2}{3}.\sqrt{(x+1)^{3}}$
$\Rightarrow A+B=-\frac{2}{5}.\sqrt{x^{5}}+\frac{2}{5}.\sqrt{(x+1)^{5}}-\frac{2}{3}.\sqrt{(x+1)^{3}}$