Mẫu $=\sqrt 2 \bigg ( 1+\sin (x+\dfrac{\pi}{4}) \bigg )=\sqrt 2 \bigg [ \sin (\dfrac{x}{2}+\dfrac{\pi}{8})+\cos (\dfrac{x}{2}+\dfrac{\pi}{8}) \bigg ]^2$
$=2\sqrt 2 \sin^2 (\dfrac{x}{2}+\dfrac{3\pi}{8})$
$I=\dfrac{1}{2\sqrt 2}\int \dfrac{1}{\sin^2 (\dfrac{x}{2}+\dfrac{3\pi}{8})}dx=\dfrac{1}{\sqrt 2}\int \dfrac{1}{\sin^2 (\dfrac{x}{2}+\dfrac{3\pi}{8})}d(\dfrac{x}{2}+\dfrac{3\pi}{8})=-\dfrac{1}{\sqrt 2}\cot (\dfrac{x}{2}+\dfrac{3\pi}{8})+C$