Chapter 5 - The Integral - 5.7 Substitution Method - Exercises - Page 276: 57

$$2 \tan (\sqrt{x})+C$$

Given $$\int \frac{\sec ^{2}(\sqrt{x})}{\sqrt{x}} d x $$ Let $$u=\sqrt{x} \ \ \ \ \Rightarrow \ \ \ du =\frac{1}{2\sqrt{x}} dx $$ \begin{aligned} \int \frac{\sec ^{2}(\sqrt{x})}{\sqrt{x}} d x &=2 \int \sec ^{2} u d u \\ &=2 \tan u+C\\ &= 2 \tan (\sqrt{x})+C \end{aligned}