integration question

Question

Transform the antiderivative problem

∫sec x dx

using the substitution u = sec x + tan x.

Your final answer should be a new antiderivative problem written in terms of u.

________

Answer 1

integral of secx = ln(secx +tanx) + c, where c is an unknown constant

integral of secx = ln|u| + c

u = secx + tanx

du = (sectan + sec^2)dx

secx = secx(secx +tanx )/(secx+tanx)

= (secxtanx + sec^2(x)/(secx +tanx)

= du/u

integral of secx = du/u

= ln|u| + c where c = an unknown constant