I'm working this problem:

Question

∫ [1-tan^2 (x)] / [sec^2 (x)] dx
∫(1/secx)-[(sin^2x/cos^2x)/(1/cosx)
∫cosx-sinx(sinx/cosx)
∫cosx-∫sin^2(x)/cosx
sinx-∫(1-cos^2(x))/cosx
sinx-∫(1/cosx)-cosx
sinx-∫secx-∫cosx
sinx-sinx-∫secx
=-ln |secxtanx|+C

Can someone just verify that I did everything correctly?

Answer 1

I think your 2nd line is bogus, and the brackets are unbalanced.

(1-tan^2)/sec^2
= 1/sec^2 - tan^2/sec^2
= cos^2 - sin^2
= cos(2x)

integral of cos(2x) = 1/2 sin(2x)