Chapter 5 - The Integral - 5.7 Substitution Method - Exercises - Page 276: 78

$$3$$

Given $$ \int_{10}^{17}(x-9)^{-\frac{2}{3}} d x$$ Let $$ u= x-9 \ \ \ \Rightarrow \ \ \ du =dx$$ At $$ x= 10\to u= 1, \ \ x= 17\to u= 8$$ Then \begin{aligned} \int_{10}^{17}(x-9)^{-\frac{2}{3}} d x &=\int_{1}^{8} u^{-\frac{2}{3}} d x \\ &=\left.\frac{u^{-\frac{2}{3}+1}}{-\frac{2}{3}+1}\right|_{1} ^{8} \\ &=3\left(8^{1/3}-1\right) \\ &=3(2-1) \\ &=3 \end{aligned}