Chapter 8 - Techniques of Integration - Chapter Review Exercises - Page 460: 9

$$ 2\tan^{-1}(\sqrt{x})+C$$

Given $$\int \frac{d x}{x^{3 / 2}+x^{1 / 2}}$$ Let $$ u^2 = x\ \ \ \ \ \ \ \ \ \ \ 2udu=dx$$ Then \begin{align*} \int \frac{d x}{x^{3 / 2}+x^{1 / 2}}&=\int \frac{2udu}{u^3+u}\\ &= \int \frac{2du}{u^2+1}\\ &= 2\tan^{-1}(u)+C\\ &= 2\tan^{-1}(\sqrt{x})+C \end{align*}