How do you find the antiderivative of #1/(1+cosx)#?

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Answer 1 · 2016-07-11T20:12:07

#=tan (x/2) + C#

#int dx qquad 1/(1 + cos x)#

double angle formula #cos 2A = 2 cos^2 A -1 #

so #cos x = 2 cos^2 (x/2) -1 #

#impliesint dx qquad 1/2 sec^2 (x/2)#

we know that #d/(du) tan u = sec^2 u# so #d/(du) tan (u/2) = sec^2 (u/2) * 1/2#

so we can play with that idea and FTC to get

#tan (x/2) + C#