Chapter 5 - The Integral - 5.3 The Indefinite Integral - Exercises - Page 253: 63

$f(x)=2x^3+x+2$

$f’’(x)=12x,$ $f’(0)=1,$ $f(0)=2$ $f’(x)=\int12xdx$ $f’(x)=6x^2+C_1$ Substitute in $f’(0)=1$ to solve for $C_1$ $1=0+C_1$ $C_1=1$ $f(x)=\int(6x^2+1)dx$ $f(x)=2x^3+x+C_2$ Substitute in $f(0)=2$ to solve for $C_2$ $2=0+0+C_2$ $C_2=2$ $f(x)=2x^3+x+2$