Integral of sec x tan x: Formula, Derivation and Examples

The integral of \sec x\tan x is \sec x + C. It is the reverse of one of the six basic trig derivatives, so it is worth knowing on sight.

Graph of sec x tan x and its antiderivative

\sec x\tan x is an odd function, so its antiderivative \sec x is even.

Derivation

1. Recall the derivative of sec x: \frac{d}{dx}\sec x = \sec x\tan x.
2. Reverse it: \sec x is an antiderivative of \sec x\tan x.
3. Add C.

If you want to see where the derivative comes from, write \sec x = (\cos x)^{-1} and use the chain rule:

Check by differentiating

Spotting it in disguise

The integral does not always arrive labelled. Each of these is \sec x\tan x:

You can also solve the first row directly with u = \cos x: \int\frac{\sin x}{\cos^2 x}\,dx = -\int u^{-2}\,du = u^{-1} + C = \sec x + C.

Definite integrals

Over a symmetric interval like \left[-\frac{\pi}{4}, \frac{\pi}{4}\right] the integral is 0, because the integrand is odd.

Substitution variants

Example 1. \int\sec(5x)\tan(5x)\,dx = \frac{1}{5}\sec(5x) + C.

Example 2: powers of sec. With u = \sec x, du = \sec x\tan x\,dx:

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

Example 4: the cosecant partner. \int\csc x\cot x\,dx = -\csc x + C, since \frac{d}{dx}\csc x = -\csc x\cot x.

Common mistakes

1. Integrating each factor separately. \int\sec x\tan x\,dx is not \ln\lvert\sec x + \tan x\rvert\cdot\ln\lvert\sec x\rvert. Integrals do not split over products.
2. Answering \tan x. That is the integral of \sec^2 x. See the integral of sec² x.
3. Answering \csc x for the cosecant version. It needs a minus sign: -\csc x + C.
4. Missing the disguise. Always try rewriting \frac{\sin x}{\cos^2 x} before reaching for a harder method.

Practise on trig integral problems.