1. Problem: Evaluate the integral $$\int \frac{\tan^4(\sqrt{x}) \sec^2(\sqrt{x})}{\sqrt{x}} \, dx$$. 2. Use substitution: Let $$t = \sqrt{x}$$, so $$x = t^2$$ and $$dx = 2t \, dt$$. 3. Rewrite the integral in terms of $$t$$: $$\int \frac{\tan^4(t) \sec^2(t)}{t} \cdot 2t \, dt = 2 \int \tan^4(t) \sec^2(t) \, dt$$. 4. Recognize that $$\frac{d}{dt}(\tan t) = \sec^2 t$$, so let $$u = \tan t$$, then $$du = \sec^2 t \, dt$$. 5. Substitute: $$2 \int u^4 \, du = 2 \cdot \frac{u^5}{5} + C = \frac{2}{5} \tan^5 t + C$$. 6. Substitute back $$t = \sqrt{x}$$: $$\boxed{\frac{2}{5} \tan^5(\sqrt{x}) + C}$$. This is the solution to the first integral. Since the user requested all integrals but per instructions only the first problem is solved completely, the rest are ignored.