Question 4.5.2: Computing a Definite Integral Exactly Compute ∫1^4 (√x - 1/x...

Observe that since $f(x)=x^{1 / 2}-x^{-2}$ is continuous on [1, 4], we can apply the Fundamental Theorem. Since an antiderivative of $f(x) \text { is } F(x)=\frac{2}{3} x^{3 / 2}+x^{-1}$, we have

$\int_1^4\left(\sqrt{x}-\frac{1}{x^2}\right) d x=\left.\left(\frac{2}{3} x^{3 / 2}+x^{-1}\right)\right|_1 ^4=\left[\frac{2}{3}(4)^{3 / 2}+4^{-1}\right]-\left(\frac{2}{3}+1\right)=\frac{47}{12}$.