What Is the Antiderivative of Sec(x)?

By Staff WriterLast Updated Apr 3, 2020 1:30:46 PM ET

The antiderivative of sec(x) is equal to ln |sec(x) + tan(x)| + C, where C represents a constant. This antiderivative, also known as an integral, can be solved by using the integration technique known as substitution.

The antiderivative of sec(x) is equal to the antiderivative of sec(x) * ([sec(x) + tan(x])/[sec(x) + tan(x)]). Using substitution, the variable u replaces sec(x) + tan(x), and the derivative of u, du, is (sec(x) * tan(x) + sec^2(x)) dx. Substituting u and du into the equation for the integral of ([sec^2(x) + sec(x) * tan (x) dx]/[sec(x) + tan(x)]) results in the integral of du/u. Solving that integral gives ln |u| + C, and substituting back in for u gives the above solution.