Indefinite Integration of tan x

Indefinite Integration of tan x

How to find the indefinite integration of tan x: example and its solution.

Example

Find the given indefinite integral. The integral of (tan x) dx

Quotient identities

Set cos x = t.

Then -sin x dx = dt,
sin x dx = -dt.

Derivative of cos x

Substitute [cos x] with [t].
And substitute [sin x dx] with [(-1)dt].

Then (given) = ∫ 1/t (-1) dt.

Integration by substitution (Part 1)

-∫ 1/t dt = -ln |t| + C



Indefinite integration of 1/x

Put cos x in t.

Then (given) = -ln |cos x| + C.

So ∫ tan x dx = -ln |cos x| + C.