Integration

I(m,n)=\int cos^{m}xsin(nx)dx \\=cos^{m}x(-\frac{cos(nx)}{n})-\int mcos^{n-1}x(-sinx)(\frac{-cox(nx)}{n})dx \\=-\frac{cos^{m}xcos(nx)}{n}-\frac{m}{n}\int cos^{m-1}xcos(nx)sinxdx \\=-\frac{cos^{m}xcos(nx)}{n}-\frac{m}{n}\int cos^{m-1}x[sin(nx)cosx-sin(n-1)x]dx \\=-\frac{cos^{m}xcos(nx)}{n}-\frac{m}{n}\int [cos^{m}xsin(nx)-cos^{m-1}sin(n-1)x]dx \\=-\frac{cos^{m}xcos(nx)}{n}-\frac{m}{n}[I(m,n)-I(m-1,n-1) \\I(m,n)=-\frac{cos^{m}xcos(nx)}{m+n}-\frac{m}{m+n}I(m-1,n-1)

or another method is

I(m,n)=∫(e^ix+e^-ix2)^m(e^inx-e^-inx2i)

Apply by-parts taking first as u and second as v