Chapter 5 - The Integral - 5.7 Substitution Method - Exercises - Page 275: 25

$$\frac{2}{3}\sin (x^{3/2}) +c$$

Given $$\int x^{1 / 2} \cos \left(x^{3 / 2}\right) d x$$ Let $$ u= x^{3/2}\ \ \ \Rightarrow \ \ \ du=\frac{3}{2}x^{1/2}dx$$ Then \begin{align*} \int x^{1 / 2} \cos \left(x^{3 / 2}\right) d x&= \int \frac{2}{3} \cos \left(u\right)du\\ &= \frac{2}{3}\sin u +c\\ &=\frac{2}{3}\sin (x^{3/2}) +c \end{align*}