What’s the integral of int tanx / (cosx)^2 dx?

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##int tanx / (cosx)^2 dx = 1/2 sec^2x +C## (Or, equivalently ##1/2tan^2x +C##, depending on system used.)

Method 1

##tanx/cos^2x = sinx/cosx 1/cos^2x = sinx (cosx)^-3##

Integrate by superabundance delay ##u=cosx##.

Method 2

##tanx/cos^2x = tanx seec^2x = (secx)(secxtanx)##

Integrate by superabundance delay ##u=secx##.

Method 3

##tanx/cos^2x = tanx seec^2x##

Integrate by superabundance delay ##u=tanx##.

This system gets the antiderivative in the shape ##1/2tan^2x + C##

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