[tex] \int\limits{\cos^4{(3x)}} \, dx = \int\limits{(\cos^2{(3x)})(\cos^2{(3x)})} \, dx \\=\int\limits{\frac{1}{2}(1+\cos{(2(3x))})\frac{1}{2}(1+\cos{(2(3x))})} \, dx \\=\frac{1}{4}\int\limits{(1+\cos{(6x)})(1+\cos{(6x)})} \, dx =\frac{1}{4}\int\limits{(1+2\cos{(6x)}+\cos^2{6x})} \, dx\\=\frac{1}{4}\int\limits{(1+2\cos{(6x)}+\frac{1}{2}(1+\cos{(2(6x}))} \, dx \\=\frac{1}{8}\int\limits{(2+4\cos{(6x)}+1+\cos{(12x}))} \, dx \\=\frac{1}{8}\int\limits{(3+4\cos{(6x)}+\cos{(12x}))} \, dx[/tex]
[tex]=\frac{1}{8} (3x+\frac{4}{6}\sin{(6x)}+\frac{1}{12}\sin{(12x}))=\frac{1}{96} (36x+8\sin{(6x)}+\sin{(12x}))[/tex]
