So I'm pretty much stuck with an integral problem. It was on the form (1 + (2t)^2)^0,5, using trig substitution I got to the integral 1/2(sec x)^3. Continuing with integration by parts, I finally got the solution (INT)1/2(sec x)^3 = 1/4(sec x)(tan x) + 1/4 ln |sec x + tan x| which should be correct. However, when I try to substitute back, using t = 1/2 tan x, my answer does not equal the answer in my book. And I can't seem to figure out what I did wrong... I've uploaded a picture of the last part of the problem, with the answer in my book below my answer.

Any help would be greatly appreciated!