Answer
$$\frac{1}{4}\sin {x^8} + C$$
Work Step by Step
$$\eqalign{
& \int {2{x^7}\cos {x^8}} dx \cr
& {\text{set }}u = {x^8}{\text{ then }}\frac{{du}}{{dx}} = 8{x^7},\,\,\,\,\,\,\,\,\frac{{du}}{{8{x^7}}} = dx \cr
& {\text{write the integrand in terms of }}u \cr
& \int {2{x^7}\cos {x^8}} dx = \int {2{x^7}\cos u} \left( {\frac{{du}}{{8{x^7}}}} \right) \cr
& {\text{cancel the common factors}} \cr
& = \int {\cos u} \left( {\frac{{du}}{4}} \right) \cr
& = \frac{1}{4}\int {\cos u} du \cr
& {\text{integrate by using the Basic Trigonometric integral }}\int {\cos x} dx = \sin x + C \cr
& = \frac{1}{4}\sin u + C \cr
& {\text{write in terms of }}x \cr
& = \frac{1}{4}\sin {x^8} + C \cr} $$
