Integral of tan x

June 3, 2026Mathematics

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Step-by-step solution

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To find the integral of tan x, we can rewrite tan x as (sin x)/(cos x).

Set up the integral.
∫ tan x dx = ∫ (sin x)/(cos x) dx
Use a substitution.
Let u = cos x.
Then, du = -sin x dx, which means sin x dx = -du.
Substitute u and du into the integral.
∫ (sin x)/(cos x) dx = ∫ (1)/(u) (-du) = -∫ (1)/(u) du
Integrate with respect to u.
-∫ (1)/(u) du = -ln|u| + C
Substitute back cos x for u.
-ln|u| + C = -ln|cos x| + C
Use logarithm properties to simplify.
Since -ln|cos x| = ln|(cos x)⁻¹| = ln|(1)/(cos x)| = ln|sec x|.
So, the integral is ln|sec x| + C.

The integral of tan x is ln|sec x| + C.

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