$\int \frac{2 \cos ^{4}(\sqrt{x}) \sin (\sqrt{x})}{3 \sqrt{x}} d x$

Step 1 ：Let $u = \sqrt{x}$, then $x = u^2$ and $dx = 2u du$

Step 2 ：Substitute $u$ into the integral: $\int \frac{2 \cos ^{4}(u) \sin (u)}{3 u} (2u) du$

Step 3 ：Simplify the integral: $\int 2 \cos ^{4}(u) \sin (u) du$

Step 4 ：Integrate with respect to $u$: $\frac{1}{12} \operatorname{Si}(u) + \frac{1}{8} \operatorname{Si}(3 u) + \frac{1}{24} \operatorname{Si}(5 u) + C$

Step 5 ：Substitute back $u = \sqrt{x}$: $\frac{1}{12} \operatorname{Si}(\sqrt{x}) + \frac{1}{8} \operatorname{Si}(3 \sqrt{x}) + \frac{1}{24} \operatorname{Si}(5 \sqrt{x}) + C$

Step 6 ：\(\boxed{\int \frac{2 \cos ^{4}(\sqrt{x}) \sin (\sqrt{x})}{3 \sqrt{x}} d x = \frac{1}{12} \operatorname{Si}(\sqrt{x}) + \frac{1}{8} \operatorname{Si}(3 \sqrt{x}) + \frac{1}{24} \operatorname{Si}(5 \sqrt{x}) + C}\)