Just do simple math : #int_0^(pi/4)(1+cos^2(x))/cos^2(x) dx #
#=int_0^(pi/4)1/cos^2(x)+1 dx#
#=(tan(x)+x)~|_0^(pi/4)#
#=1+pi/4#
EDIT : Here the "Funny" answer !
#int_0^(pi/4)(1+cos^2(x))/cos^2(x)dx = int_0^(pi/4)(2+cos^2(x)-1)/cos^2(x)dx#
Factorize : #int_0^(pi/4)(-(1-cos^2(x))+2)/cos^2(x)dx = int_0^(pi/4)(-sin^2(x)+2)/cos^2(x)dx#
#=-int_0^(pi/4)sin^2(x)/cos^2(x) dx+2int_0^(pi/4)1/cos^2(x) dx#
#=-int_0^(pi/4)1+tan^2(x)-1 dx + 2int_0^(pi/4)1/cos^2(x)dx#
#=-(tan(x)-x)~|_0^(pi/4)+2(tan(x))~|_0^(pi/4)#
#=-1+pi/4+2 = 1+pi/4#
