Answer
$$g\left( x \right) = \frac{{7{x^8}}}{8} - \frac{{{x^2}}}{2} + \frac{{13}}{8}$$
Work Step by Step
$$\eqalign{
& g'\left( x \right) = 7x\left( {{x^6} - \frac{1}{7}} \right) \cr
& {\text{multiply}} \cr
& g'\left( x \right) = 7{x^7} - x \cr
& g\left( x \right) = \int {g'\left( x \right)} dx \cr
& then \cr
& g\left( x \right) = \int {\left( {7{x^7} - x} \right)} dx \cr
& find{\text{ the general solution}} \cr
& g\left( x \right) = \frac{{7{x^8}}}{8} - \frac{{{x^2}}}{2} + C \cr
& {\text{using the initial condition }}g\left( 1 \right) = 2 \cr
& 2 = \frac{{7{{\left( 1 \right)}^8}}}{8} - \frac{{{{\left( 1 \right)}^2}}}{2} + C \cr
& 2 = \frac{3}{8} + C \cr
& C = \frac{{13}}{8} \cr
& {\text{the solution to the initial value problem is}} \cr
& g\left( x \right) = \frac{{7{x^8}}}{8} - \frac{{{x^2}}}{2} + \frac{{13}}{8} \cr} $$
