The antiderivative of tan(x) can be expressed as either - ln |cos(x)| + C or as ln |sec(x)| + C. In these equations, C indicates a constant, ln is the natural logarithm function, cos indicates the function cosine and sec denotes the function secant.
Both solutions for the antiderivative of tan(x) can be found by using an integration technique called substitution. In integration, the substitution technique works in the reverse manner of the chain rule for differentiation. Using substitution, the variable u is used to replace one of the terms in the integral and the derivative of u replaces the term dx in the function. After solving in terms of u, the terms from the original function are substituted back in place of u and du.