Integral of tan x

tan x = (sin x)/(cos x)

Quotient Identity

Solve this integral by substitution.

First set cos x = t.

Differentiate both sides.

The derivative of cos x is -sin x dx.
And the derivative of t is dt.

Derivative of cos x

Derivative of an Implicit Function

To make sin x dx,
change -sin x dx = dt
to sin x dx = -dt.

cos x = t
sin x dx = -dt

Put these into ∫ (sin x)/(cos x) dx.

Then ∫ 1/t (-dt).

Take the minus sign out from the integral.

Solve the integral.

Write minus.

The integral of 1/t is ln |t|.

Integral of 1/x

This is an indefinite integral.
So write +C.

cos x = t
So change t back to cos x.

So the integral of tan x is
-ln |cos x| + C.