[tex]I=\displaystyle\int_0^{\pi/4}\tan^2x\sec x\,\mathrm dx[/tex]
[tex]\displaystyle\int_0^{\pi/4}\tan^4x\sec x\,\mathrm dx=uv\bigg|_{x=0}^{x=\pi/4}-\int_0^{\pi/4}v\,\mathrm du[/tex]
where
[tex]u=\tan^3x\implies\mathrm du=3\tan^2x\sec^2x\,\mathrm dx[/tex]
[tex]\mathrm dv=\sec x\tan x\,\mathrm dx\implies v=\sec x[/tex]
[tex]\displaystyle\int_0^{\pi/4}\tan^4x\sec x\,\mathrm dx=\tan^3x\sec x\bigg|_{x=0}^{x=\pi/4}-3\int_0^{\pi/4}\tan^2x\sec^3x\,\mathrm dx[/tex]
[tex]\displaystyle\int_0^{\pi/4}\tan^4x\sec x\,\mathrm dx=\sqrt2-3\int_0^{\pi/4}\tan^2x(1+\tan^2x)\sec x\,\mathrm dx[/tex]
[tex]\displaystyle\int_0^{\pi/4}\tan^4x\sec x\,\mathrm dx=\sqrt2-3I-3\int_0^{\pi/4}\tan^4x\sec x\,\mathrm dx[/tex]
[tex]4\displaystyle\int_0^{\pi/4}\tan^4x\sec x\,\mathrm dx=\sqrt2-3I[/tex]
[tex]\displaystyle\int_0^{\pi/4}\tan^4x\sec x\,\mathrm dx=\dfrac{\sqrt2-3I}4[/tex]
