Answer
$$\frac{{{4^x}}}{{\ln 4}} + C$$
Work Step by Step
$$\eqalign{
& \int {{4^x}} dx \cr
& {\text{We know that }}\frac{d}{{dx}}\left[ {{a^x}} \right] = {a^x}\ln a,{\text{ then }}\int {{a^x}dx} = \frac{{{a^x}}}{{\ln a}} + C \cr
& {\text{Therefore}} \cr
& \underbrace {\int {{4^x}} dx}_{\int {{a^x}dx} } = \underbrace {\frac{{{4^x}}}{{\ln 4}} + C}_{\frac{{{a^x}}}{{\ln a}} + C} \cr
& {\text{ }} = \frac{{{4^x}}}{{\ln 4}} + C \cr} $$