AP Calculus BC

Integral of sec^3 x: The Classic Loop

Q: What is the integral of sec^3 x?

It is \frac{\sec x\tan x + \ln\left|\sec x+\tan x\right|}{2} + C.

Q: Why does the original integral reappear?

The identity \tan^{2}x = \sec^{2}x - 1 reintroduces \sec^{3}x. Solving for it algebraically is the method.

The integral of secant cubed x is secant x tangent x plus the natural log of the absolute value of secant x plus tangent x, all over 2, plus C. Splitting off one secant squared for dv and using parts makes the original integral reappear, so you solve for it.

\int \sec^{3} x\,dx = \frac{\sec x\tan x + \ln\left|\sec x + \tan x\right|}{2} + C

Split, then solve for the integral

Write $\sec^{3}x = \sec x\cdot\sec^{2}x$ , take $u = \sec x$ and $dv = \sec^{2}x\,dx$ , so $v = \tan x$ . Using $\tan^{2}x = \sec^{2}x - 1$ brings the original integral back.

\int \sec^{3}x\,dx = \sec x\tan x - \int \sec^{3}x\,dx + \int \sec x\,dx

Adding the integral to both sides and halving gives the answer, with the secant integral supplying the logarithm.

Why it is famous

It combines three separate ideas: parts, a Pythagorean identity, and solving for the unknown integral. Trigonometric substitution problems reduce to it constantly, which is why it is worth recognising even though AP does not test that substitution.

Common mistakes

Forgetting the $\frac{1}{2}$ after moving the integral across.
Splitting as $\sec^{2}x\cdot\sec x$ with the roles swapped, which does not produce a workable $v$ .

Every answer on this page is machine checked

An automated test differentiates the antiderivative above and confirms it returns the integrand. A wrong sign or a missing factor fails the build, so it cannot reach you.

Frequently asked questions

What is the integral of sec^3 x?

It is $\frac{\sec x\tan x + \ln\left|\sec x+\tan x\right|}{2} + C$ .

Why does the original integral reappear?

The identity $\tan^{2}x = \sec^{2}x - 1$ reintroduces $\sec^{3}x$ . Solving for it algebraically is the method.