1. **State the problem:** Evaluate the integral $$\int \cos x \sqrt{\sin x} \, dx$$ using substitution. 2. **Identify substitution:** Let $$u = \sin x$$, then $$du = \cos x \, dx$$. 3. **Rewrite the integral:** Substitute $$u$$ and $$du$$ into the integral: $$\int \cos x \sqrt{\sin x} \, dx = \int \sqrt{u} \, du = \int u^{\frac{1}{2}} \, du$$. 4. **Integrate:** Use the power rule for integration: $$\int u^{\frac{1}{2}} \, du = \frac{u^{\frac{3}{2}}}{\frac{3}{2}} + C = \frac{2}{3} u^{\frac{3}{2}} + C$$. 5. **Back-substitute:** Replace $$u$$ with $$\sin x$$: $$\frac{2}{3} (\sin x)^{\frac{3}{2}} + C$$. **Final answer:** $$\int \cos x \sqrt{\sin x} \, dx = \frac{2}{3} (\sin x)^{\frac{3}{2}} + C$$