Answer
$$\sin 4w + \cos 3w + C$$
Work Step by Step
$$\eqalign{
& \int {\left( {4\cos 4w - 3\sin 3w} \right)dw} \cr
& {\text{split the integrand}} \cr
& = \int {4\cos 4wdw} - \int {3\sin 3wdw} \cr
& = 4\int {\cos 4wdw} - 3\int {\sin 3wdw} \cr
& {\text{use the formula for indefinite integrals of trigonometric functions}} \cr
& = 4\left( {\frac{1}{4}\sin 4w} \right) - 3\left( { - \frac{1}{3}\cos 3w} \right) + C \cr
& {\text{simplify}} \cr
& = \sin 4w + \cos 3w + C \cr
& {\text{check by differentiation}} \cr
& {\text{ = }}\frac{d}{{dw}}\left( {\sin 4w + \cos 3w + C} \right) \cr
& {\text{ = }}\frac{d}{{dw}}\left( {\sin 4w} \right) + \frac{d}{{dw}}\left( {\cos 3w + C} \right) + {\text{ = }}\frac{d}{{dw}}\left( C \right) \cr
& {\text{ = }}4\cos 4w - 3\sin 3w + 0 \cr
& = 4\cos 4w - 3\sin 3w \cr} $$
