1. **State the problem:** We want to evaluate the integral $$\int \tan^5 x \sec^2 x \, dx.$$\n\n2. **Recall the formula and substitution:** Notice that the derivative of $\tan x$ is $\sec^2 x$. This suggests using substitution with $u = \tan x$.\n\n3. **Substitution:** Let $u = \tan x$, then $du = \sec^2 x \, dx$. The integral becomes $$\int u^5 \, du.$$\n\n4. **Integrate:** $$\int u^5 \, du = \frac{u^6}{6} + C.$$\n\n5. **Back-substitute:** Replace $u$ with $\tan x$ to get the final answer: $$\frac{\tan^6 x}{6} + C.$$