Step 1 To solve the integral. Step 2 To solve int (1)/(cos^(4)x)dx Using sec x=(1)/(cos x), we have int (1)/(cos^(4)x)dx=int sec^(4)x dx =int sec^(2)x(tan^(2)x+1)dx [using, sec^(2)x=tan^(2)x+1] we substitute u=tan x -> (du)/(dx)=sec^(2)x -> dx=(1)/(sec^(2)x)du Thus, int (1)/(cos^(4)x)dx=int (u^(2)+1)du=(u^(3))/(3)+u Undo substitution u=tan x; we have int (1)/(cos^(4)x)dx=(tan^(3)x)/(3)+tan x+C where C is constant of integration.