How do you integrate tan (x)?

We need to find the integral of tan x

Substitute in terms of sin’s and cos’s; Use Substitution.

∫ tan x dx = ∫sin x / cos x dx
Substitute u = cos x.
we get
du = – sin x dx

substitute du=-sin x, u=cos x

∫ sin x/cos x dx

= – ∫
(-1) sin x dx / cos x

= – ∫ du / u
On solving it we get

= – ln |u| + C

on back substitution u = cos x we get

= – ln |cos x| + C

∫tan x =- ln |cos x| + C

Answer

∫tan x =- ln |cos x| + C

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