Evaluating a definite integral combining power and reciprocal power rules

Rewrite the integrand as $x^{1/2} + x^{-2}$. Using the power rule for antiderivatives, $\int x^{1/2}dx = \frac{2}{3}x^{3/2}$ and $\int x^{-2}dx = -x^{-1}$. So an antiderivative is $F(x) = \frac{2}{3}x^{3/2} - \frac{1}{x}$.

By FTC Part 2, the integral equals $F(9) - F(4)$. Compute $F(9) = \frac{2}{3}(27) - \frac{1}{9} = 18 - \frac{1}{9} = \frac{161}{9}$, and $F(4) = \frac{2}{3}(8) - \frac{1}{4} = \frac{16}{3} - \frac{1}{4} = \frac{64}{12} - \frac{3}{12} = \frac{61}{12}$. Subtracting, $\frac{161}{9} - \frac{61}{12} = \frac{644}{36} - \frac{183}{36} = \frac{461}{36}$.

The answer is F of four. By FTC Part 2, the definite integral from four to nine equals the antiderivative evaluated at the upper limit minus the antiderivative evaluated at the lower limit, that is F of nine minus F of four. In part a this gave the expression F of nine minus F of four, which was computed as eighteen minus one ninth minus the value at four.