Evaluating [latex]{\displaystyle\int}{\sec}^{n}xdx[/latex] for values of [latex]n[/latex] where [latex]n[/latex] is odd requires integration by parts. In addition, we must also know the value of [latex]{\displaystyle\int}{\sec}^{n - 2}xdx[/latex] to evaluate [latex]{\displaystyle\int}{\sec}^{n}xdx[/latex]. The evaluation of [latex]{\displaystyle\int}{\tan}^{n}xdx[/latex] also requires being able to integrate [latex]{\displaystyle\int}{\tan}^{n - 2}xdx[/latex]. To make the process easier, we can derive and apply the following power reduction formulas. These rules allow us to replace the integral of a power of [latex]\sec{x}[/latex] or [latex]\tan{x}[/latex] with the integral of a lower power of [latex]\sec{x}[/latex] or [latex]\tan{x}[/latex].
reduction formulas for [latex]{\displaystyle\int}{\sec}^{n}xdx[/latex] and [latex]{\displaystyle\int}{\tan}^{n}xdx[/latex]