Answer
$$\frac{{{{10}^x}}}{{\ln 10}} + 4\ln \left| x \right| - \cos x + C$$
Work Step by Step
$$\eqalign{
& \int {\left( {{{10}^x} + \frac{4}{x} + \sin x} \right)} dx \cr
& {\text{sum rule for derivatives}} \cr
& = \int {{{10}^x}} dx + \int {\frac{4}{x}} dx + \int {\sin x} dx \cr
& \cr
& {\text{Integrate using the rules of integration}} \cr
& \int {{a^x}} dx = \frac{{{a^x}}}{{\ln a}} + C,\,\,\,\int {\frac{1}{x}} dx = \ln \left| x \right| + C{\text{ and }}\int {\sin x} dx = - \cos x + C \cr
& \cr
& {\text{then}} \cr
& = \frac{{{{10}^x}}}{{\ln 10}} + 4\ln \left| x \right| - \cos x + C \cr} $$
