∫π[√2-tan(4x)sec(4x)]²dx
=π∫[2-2√2tan(4x)sec(4x)+tan²(4x)sec²(4x)]dx
=π[2∫dx-2√2∫tan(4x)sec(4x)dx+∫tan²(4x)sec²(4x)dx]
=π[2x-(√2/2)∫tan(4x)sec(4x)d(4x)+(1/4)∫tan²(4x)d(tan(4x))]
=π[2x-(√2/2)sec(4x)+(1/4)*(1/3)tan³(4x)]+C
=π[2x-(√2/2)sec(4x)+(1/12)tan³(4x)]+C
C为任意常数