Tìm nguyên hàm của hàm số sau: ∫dx/cosx.cos(x+π4)

Question

\(\int \frac{{\rm d}x}{\cos x.\cos \left(x+\frac{\pi }{4} \right)}\)

Answer 1

 Ta có: \(I=\int \frac{{\rm d}x}{\cos x.\cos \left(x+\frac{\pi }{4} \right)} =\int \frac{{\rm d}x}{\cos x.\left(\frac{\sqrt{2} }{2} \cos x-\frac{\sqrt{2} }{2} \sin x\right)} =\int \frac{\sqrt{2} \, {\rm d}x}{\cos ^{2} x.\left(1-\tan x\right)}\).

 Đặt \(t=1-\tan x\Rightarrow {\rm d}t=-\frac{1}{\cos ^{2} x} {\rm d}x\Rightarrow \frac{1}{\cos ^{2} x} {\rm d}x=-{\rm d}t.\)

 Suy ra: \(I=-\sqrt{2} \int \frac{{\rm d}t}{t} =-\sqrt{2} \ln \left|t\right|+C.\)

 Vậy \(I=-\sqrt{2} \ln \left|1-\tan x\right|+C.\)