Ta biến đổi f(x) về dạng :
\(f\left(x\right)=\frac{\sin x.\sin\left(x+\frac{\pi}{4}\right)+\cos x.\cos\left(x+\frac{\pi}{4}\right)}{\cos x.\cos\left(x+\frac{\pi}{4}\right)}-1=\frac{\cos\frac{\pi}{4}}{\cos x.\cos\left(x+\frac{\pi}{4}\right)}-1\)
\(\Rightarrow F\left(x\right)=\frac{\sqrt{2}}{2}\int\frac{dx}{\cos x.\cos\left(x+\frac{\pi}{4}\right)}dx-\int dx=\frac{\sqrt{2}}{2}\int\frac{dx}{\cos x.\cos\left(x+\frac{\pi}{4}\right)}dx-x\left(1\right)\)
Để tính \(J=\int\frac{dx}{\cos x.\cos\left(x+\frac{\pi}{4}\right)}dx\)
Ta có \(\int\frac{dx}{\cos x.\cos\left(x+\frac{\pi}{4}\right)}dx=\sqrt{2}\int\frac{1}{\cos x.\left(\cos x-\sin x\right)}dx=\sqrt{2}\int\frac{1}{\left(1-\tan x\right)}.\frac{1}{\cos^2x}dx\)
\(=-\sqrt{2}\int\frac{d\left(1-\tan x\right)}{1-\tan x}=\sqrt{2}\ln\left|1-\tan x\right|+C\)