Đặt \(\left\{\begin{array}{l} {u=x} \\ {{\rm d}v=\frac{1}{\cos ^{2} x} {\rm d}x} \end{array}\right. \Rightarrow \left\{\begin{array}{l} {{\rm d}u={\rm d}x} \\ {v=\tan x} \end{array}\right. \)
Khi đó \(\int \frac{x}{\cos ^{2} x} {\rm d}x =x\tan x-\int \frac{\sin x}{\cos x} {\rm d}x =x\tan x+\int \frac{1}{\cos x} {\rm d}\left(\cos x\right)=\, x.\tan x+\ln \left|\cos x\right|+C.\)