Answer
$$\int {{u^n}\cos u} du = {u^n}\sin u - n\int {{u^{n - 1}}\sin u} du$$
Work Step by Step
$$\eqalign{
& \int {{u^n}\cos u} du \cr
& {\text{Use integration by parts}} \cr
& t = {u^n},{\text{ }}dt = n{u^{n - 1}}du \cr
& dv = \cos udu,{\text{ }}v = \sin u \cr
& \int {tdv} = tv - \int {vdt} \cr
& {\text{Substituting}} \cr
& \int {{u^n}\cos u} du = \left( {{u^n}} \right)\left( {\sin u} \right) - \int {\sin u} \left( {n{u^{n - 1}}} \right)du \cr
& \int {{u^n}\cos u} du = {u^n}\sin u - \int {n{u^{n - 1}}\sin u} du \cr
& \int {{u^n}\cos u} du = {u^n}\sin u - n\int {{u^{n - 1}}\sin u} du \cr} $$
