The process of integrating the tangent function involves understanding its relation to other trigonometric identities and simplifying the expression. The tangent function, represented as tan(x), is commonly encountered in calculus problems, especially in the context of integration.

One effective way to approach the integral of tan(x) is by rewriting it using the identity:

  • tan(x) = sin(x) / cos(x)

This allows for a more straightforward integration process, which can be solved using substitution methods. Here’s a step-by-step breakdown:

  1. Set u = cos(x), then du = -sin(x) dx.
  2. Substitute into the integral, yielding -∫(du/u).
  3. Finally, the result of this integration is -ln|u| + C.

Important note: After substitution, always back-substitute u = cos(x) to get the final answer in terms of x.

The integral of tan(x) is thus -ln|cos(x)| + C, where C is the constant of integration.