On the simplification of trigonometric function in higher mathematics integral, how to change the place with red strokes and double lines?

∫secx = 1/cosx = 1/sin(π/2+x)= CSC(x+π/2)； dx = d(x+π/2)；

∴∫sec xdx =∫CSC(x+π/2)d(x+π/2)=∫CSC udu =∫du/sinu =∫du/[2 sin(u/2)cos(u/。

=∫d(u/2)/[tan(u/2)cos？ (u/2)]=∫[ sec? (u/2)d(u/2)]/tan(u/2)=∫d[tan(u/2)]/tan(u/2)=ln∣tan(u/2)∣+c；

tan(u/2)=[sin(u/2)]/[coa(u/2)]=[2 sin？ (u/2)]/[2 sin(u/2)cos(u/2)]=[2 sin？ (u/2)]/ (sine)

=( 1-cosu)/sinu=cscu-cotu

So ∫ secxdx = ln (cscu-cotu)+c = ln ∣ CSC (x+π/2)-cot (x+π/2) ∣+c = ln ∣ secx+tanx ∣+C.

CSC(x+π/2)= secx； Cot(x+π/2)=-tanx, which is the most basic inductive formula of trigonometric function.

This is an integral formula that can be directly selected.