Chapter 5 - Section 5.4 - The Fundamental Theorem of Calculus - Exercises - Page 320: 24

$-\frac{137}{20}$

\ne\begin{aligned} \int_{1}^{8} \frac{\left(x^{1 / 3}+1\right)\left(2-x^{2 / 3}\right)}{x^{1 / 3}} d x&=\int_{1}^{8} \left(\frac{2x^{\frac{1}{3}}-x+2-x^{\frac{2}{3}}}{x^{\frac{1}{3}}}\right) d x\\ &= \int_{1}^{8} \left(2-x^{\frac{2}{3}}+2x^{\frac{-1}{3}}-x^{\frac{1}{3}}\right) d x\\ &= \left(2x- \frac{3}{5}x^{5/3} + 3x^{2/3}- \frac{3}{4}x^{4/3}\right)\bigg|_1^8\\ &= \left(16-\frac{96}{5}+12-12\right)- \left(2- \frac{3}{5} + 3- \frac{3}{4}\right)\\ &=-\frac{137}{20}\end{aligned}