Answer
$$\frac{{{{\cos }^3}2\pi t\sin 2\pi t}}{\pi } + \frac{{3\cos 2\pi t\sin 2\pi t}}{{2\pi }} + 3t + C $$
Work Step by Step
$$\eqalign{
& \int {8{{\cos }^4}2\pi t} dt \cr
& = 8\int {{{\cos }^4}2\pi t} dt \cr
& {\text{Use the reduction formula }}\left( {68} \right) \cr
& \,\,\,\int {{{\cos }^n}ax} dx = \frac{{{{\cos }^{n - 1}}ax\sin ax}}{{na}} + \frac{{n - 1}}{n}\int {co{s^{n - 2}}ax} dx \cr
& {\text{let }}n = 4{\text{ and }}a = 2\pi,\,\,\,x = t \cr
& \,\,\,8\int {{{\cos }^4}2\pi t} dt = 8\left( {\frac{{{{\cos }^{4 - 1}}2\pi t\sin 2\pi t}}{{4\left( {2\pi } \right)}}} \right) + 8\left( {\frac{{4 - 1}}{4}} \right)\int {{{\cos }^{4 - 2}}2\pi t} dt \cr
& \,\,\,\int {{{\cos }^4}2\pi t} dt = \frac{{{{\cos }^3}2\pi t\sin 2\pi t}}{\pi } + 6\int {{{\cos }^2}2\pi t} dt \cr
& \cr
& {\text{Integrate }}\int {{{\cos }^2}2\pi t} dt\cr
& {\text{using the reduction formula with }}\cr
& n = 2{\text{ and }}a = 2\pi \cr
& \,\int {{{\cos }^4}2\pi t} dt = \frac{{{{\cos }^3}2\pi t\sin 2\pi t}}{\pi } + 6\left( {\frac{{{{\cos }^{2 - 1}}2\pi t\sin 2\pi t}}{{4\pi }} + \frac{{2 - 1}}{2}\int {co{s^{2 - 2}}2\pi t} dt} \right) \cr
& \,\int {{{\cos }^4}2\pi t} dt = \frac{{{{\cos }^3}2\pi t\sin 2\pi t}}{\pi } + \frac{{3\cos 2\pi t\sin 2\pi t}}{{2\pi }} + 3\int {dt} \cr
& \,\int {{{\cos }^4}2\pi t} dt = \frac{{{{\cos }^3}2\pi t\sin 2\pi t}}{\pi } + \frac{{3\cos 2\pi t\sin 2\pi t}}{{2\pi }} + 3t + C \cr} $$
