What is the integral of sec x?

by

The integral of secant, denoted as ∫ sec(x) dx, can be derived through a clever manipulation involving trigonometric identities and substitution. The result is a bit surprising and showcases the beauty of calculus.

To find the integral of sec(x), we can use the following identity:

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

Here’s a brief walkthrough of how we arrive at this result:

  1. Rewrite sec(x): Start with the integral:
    2. ∫ sec(x) dx
  2. Multiply and Divide: Multiply by (sec(x) + tan(x)) / (sec(x) + tan(x)) to facilitate the substitution:
    3. ∫ sec(x) dx = ∫ rac{sec(x)(sec(x) + tan(x))}{sec(x) + tan(x)} dx
  3. Substitution: Let u = sec(x) + tan(x). Then, the derivative du = (sec(x)tan(x) + sec^2(x)) dx simplifies our integral.
  4. Change of Variables: After substituting, the integral becomes:
    5. ∫ rac{du}{u}
  5. Integrate: The integral of 1/u is:
    6. ln |u| + C = ln |sec(x) + tan(x)| + C

Thus, the final result for the integral of sec(x) is:

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

Where C is the constant of integration. This integral showcases how trigonometric functions can lead to logarithmic results, making calculus an exciting journey!

Leave a Comment