Integral of tan² x: Formula, Derivation and Examples

The integral of \tan^2 x is \tan x - x + C. One identity turns it into two integrals you already know.

Graph of tan² x and its antiderivative

Derivation, step by step

1. Start from the Pythagorean identity \sin^2 x + \cos^2 x = 1. Divide by \cos^2 x to get \tan^2 x + 1 = \sec^2 x.
2. Rearrange: \tan^2 x = \sec^2 x - 1.
3. Substitute: \displaystyle\int\tan^2 x\,dx = \int\sec^2 x\,dx - \int 1\,dx.
4. Use the integral of sec² x, \tan x, and the integral of a constant, x.
5. Combine: \tan x - x + C.

Check by differentiating

Definite integrals

Since \tan^2 x is even, \int_{-\pi/4}^{\pi/4}\tan^2 x\,dx = 2 - \frac{\pi}{2}.

Substitution variants

Note that only the \tan term gets the factor \frac{1}{a}. The -x comes from integrating the constant 1, which has no inside function.

Example 1. \int\tan^2(3x)\,dx = \frac{1}{3}\tan(3x) - x + C.

Example 2. \int\tan^2\frac{x}{2}\,dx = 2\tan\frac{x}{2} - x + C.

Higher powers of tan

The identity \tan^2 x = \sec^2 x - 1 is the key to every power of \tan x.

Example 3: tan³ x.

The first piece is \frac{\tan^2 x}{2} with u = \tan x. The second is the integral of tan x. So

Example 4: tan² x sec² x. Here u = \tan x works directly: \int\tan^2 x\sec^2 x\,dx = \frac{\tan^3 x}{3} + C. This is the one case where \frac{\tan^3 x}{3} is the right answer, because the \sec^2 x factor is present.

Example 5: tan⁴ x. \tan^4 x = \tan^2 x(\sec^2 x - 1) = \tan^2 x\sec^2 x - \tan^2 x, so

Common mistakes

1. Answering \frac{\tan^3 x}{3}. That needs a \sec^2 x factor. Check: its derivative is \tan^2 x\sec^2 x.
2. Answering \sec^2 x or \ln^2\lvert\sec x\rvert. Neither differentiates back to \tan^2 x.
3. Getting the identity backwards. It is \sec^2 x - 1, not 1 - \sec^2 x. Test at x = \frac{\pi}{4}: \tan^2 = 1 and \sec^2 - 1 = 2 - 1 = 1.
4. Dividing -x by a in \int\tan^2(ax)\,dx.

Practise on trig integral problems.