is this legal?

inte[sqrt(144(cos²(2t) + sin²(2t)) + 25)]dt

What you are doing is u substitution. in order for your theory to be viable, you would have to do the following:

u= 2t --> du = 2 dt

so technically the answer you have is wrong since you did not compensate for the 2dt...

u was just a random variable. i didnt do u substitution, it was trig substitution/trig identity.