Integral $int frac{sin^n(x)}{cos(x)}dx$
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In one of my exercises about integration we had to solve the following integral: begin{equation} int frac{sin^n(x)}{cos^m(x)}dx end{equation} We had to do this via a recursive integral. I found: begin{equation} mathcal{K_{m,n}} = frac{sin^{n-1}(x)}{(m-1)cdotcos^{m-1}(x)}-frac{n-1}{m-1}cdotmathcal{K}_{m-2,n-2}, qquad n,mgeq2 end{equation} I know for a fact that this solution is correct because we solved this in class, but the other cases where m and/or n are not $geq$ 2, were left as an exercise for us at home. I've found a solution for every case except for the case where $m = 1$ , which makes the following integral. begin{equation} int frac{sin^n(x)}{cos(x)}dx end{equation} I've tried different things, I tried integration by parts with many different u's and v's