# \ #
# "We will use one simple elementary trig rewrite to do this." #
# "Here we go:" #
# int \ 1 / { cos^2(2x) } \ dx \ = \ int \ sec^2(2x) \ dx, #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad "as" \ \ sec(\theta) \ \ "is the reciprocal of" \ \ cos(theta);#
# \qquad \qquad \qquad \qquad \qquad \qquad = \ 1/2 int \ 2 sec^2(2x) \ dx; #
# \qquad \qquad \qquad \qquad \qquad \qquad = \ 1/2 int \ d(tan(2x)) \qquad "as"\ \ [ tan(\theta) ]' \ = \ sec^2(\theta);#
# \qquad \qquad \qquad \qquad \qquad \qquad = \ 1/2 tan(2x) + C. #
# "Thus:" #
# qquad \qquad \qquad \qquad \qquad int \ 1 / { cos^2(2x) } \ dx \ = \ 1/2 tan(2x) + C. #