Question

\(\int\frac{dx}{\sqrt{x+2}+\sqrt{2-x}}\)

Answer 1

\(\int\frac{\sqrt{x+2}-\sqrt{2-x}}{x+2-\left(2-x\right)}dx=\int\frac{\sqrt{x+2}-\sqrt{2-x}}{2x}dx=\frac{1}{2}\int\frac{\sqrt{x+2}}{x}dx-\frac{1}{2}\int\frac{\sqrt{2-x}}{x}dx=\frac{1}{2}\left(I_1+I_2\right)\)

TÍNH \(I_1=\int\frac{\sqrt{x+2}}{x}dx\)

đặt \(\sqrt{x+2}=t\Rightarrow t^2=x+2\Rightarrow x=t^2-2\Rightarrow dx=2tdt\)

thay vào \(I_1=\int\frac{\sqrt{x+2}}{x}dx=\int\frac{2t^2}{t^2-2}dt=\int2dt+\int\frac{4}{t^2-2}dt=2t+\int\frac{dt}{\left(t-\sqrt{2}\right)\left(t+\sqrt{2}\right)}=2t+\frac{1}{2\sqrt{2}}ln\left|\frac{t-\sqrt{2}}{t+\sqrt{2}}\right|+C\)

THAY \(\sqrt{x+2}=t\)

tính \(I_2\)tương tự \(I_1\) TA SUY ra đc I