What is the Integral of sec 2x?

The integral of sec 2x is a common problem in calculus. To solve it, we can use a standard integration technique.

The integral of sec 2x with respect to x is:

∫ sec 2x dx = (1/2) ln |sec 2x + tan 2x| + C

Here’s a step-by-step explanation:

1. Recognize the Integral: The integral of sec 2x is not straightforward, so we use a substitution method.
2. Substitution: Let u = 2x. Then, du = 2 dx, which means dx = du/2.
3. Rewrite the Integral: Substitute u and dx into the integral: ∫ sec u (du/2).
4. Simplify: The integral becomes (1/2) ∫ sec u du.
5. Integrate: The integral of sec u is ln |sec u + tan u|.
6. Substitute Back: Replace u with 2x to get (1/2) ln |sec 2x + tan 2x| + C, where C is the constant of integration.

This is the final result of the integral of sec 2x.