Chapter 5 - The Integral - 5.7 Substitution Method - Exercises - Page 276: 71

$$3-\sqrt{5} $$

Given $$ \int_{0}^{2} \frac{d x}{\sqrt{2 x+5}} $$ Let $$u=2 x+5 \ \ \ \ \Rightarrow \ \ \ du =2dx $$ At $x=0 \to u= 5$ and at $x=2 \to u=9 $ Then \begin{aligned} \int_{0}^{2} \frac{d x}{\sqrt{2 x+5}} &=\frac{1}{2} \int_{5}^{9} \frac{1}{\sqrt{u}} d u \\ &=\frac{1}{2} \int_{5}^{9} u^{-1 / 2} d u \\ &=\left.\frac{1}{2}\left(2 u^{1 / 2}\right)\right|_{5} ^{9} \\ &=3-\sqrt{5} \end{aligned}