1. **Problem Statement:** Evaluate the integral $$\int \sqrt{1 + \sqrt{x}} \, dx.$$\n\n2. **Substitution:** Let $$u = 1 + \sqrt{x}.$$ Then $$\sqrt{x} = u - 1$$ and $$x = (u - 1)^2.$$\n\n3. **Find $$dx$$ in terms of $$du$$:**\nSince $$\sqrt{x} = x^{1/2},$$ we have $$\frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}}.$$\nDifferentiating $$u = 1 + \sqrt{x},$$ we get $$du = \frac{1}{2\sqrt{x}} dx = \frac{1}{2(u-1)} dx,$$ so $$dx = 2(u-1) du.$$\n\n4. **Rewrite the integral:**\n$$\int \sqrt{1 + \sqrt{x}} \, dx = \int \sqrt{u} \cdot 2(u-1) du = 2 \int u^{1/2} (u - 1) du = 2 \int (u^{3/2} - u^{1/2}) du.$$\n\n5. **Integrate term-by-term:**\n$$2 \left( \int u^{3/2} du - \int u^{1/2} du \right) = 2 \left( \frac{u^{5/2}}{5/2} - \frac{u^{3/2}}{3/2} \right) + C = 2 \left( \frac{2}{5} u^{5/2} - \frac{2}{3} u^{3/2} \right) + C.$$\n\n6. **Simplify:**\n$$= 2 \left( \frac{2}{5} u^{5/2} - \frac{2}{3} u^{3/2} \right) + C = \frac{4}{5} u^{5/2} - \frac{4}{3} u^{3/2} + C.$$\n\n7. **Substitute back $$u = 1 + \sqrt{x}$$:**\n$$\int \sqrt{1 + \sqrt{x}} \, dx = \frac{4}{5} (1 + \sqrt{x})^{5/2} - \frac{4}{3} (1 + \sqrt{x})^{3/2} + C.$$\n\n**Final answer:** Option (b) $$\frac{4}{5} (1 + \sqrt{x})^{5/2} - \frac{4}{3} (1 + \sqrt{x})^{3/2} + C.$$