Answer
$F(x) = x^5 - \frac{x^{6}}{3} + 4$
Work Step by Step
$F(x) = \int 5x^4 - 2x^5dx$
$F(x) = \frac{5x^{4+1}}{4+1} - \frac{2x^{5+1}}{5+1} + C$
$F(x) = \frac{5x^{5}}{5} - \frac{2x^{6}}{6} + C$
$F(x) = x^5 - \frac{x^{6}}{3} + C$
$F(0) = 0^5 - \frac{0^{6}}{3} + C$
Given: $F(0) = 4$
$4 = 0 - 0 + C$
$C = 4$
Substitute back the value of C:
$F(x) = x^5 - \frac{x^{6}}{3} + C$
$F(x) = x^5 - \frac{x^{6}}{3} + 4$
Graph the function of $f(x)$ and $F(x)$. The graphs make sense -- for instance $f(x)=0$ when $F(x)$ is at a minimum.
