1. The problem is to find the integral $$\int \tan^2(5x) \, dx$$. 2. Recall the identity $$\tan^2(\theta) = \sec^2(\theta) - 1$$. This helps us rewrite the integral in terms of secant squared, which is easier to integrate. 3. Substitute $$\tan^2(5x) = \sec^2(5x) - 1$$ into the integral: $$\int \tan^2(5x) \, dx = \int (\sec^2(5x) - 1) \, dx = \int \sec^2(5x) \, dx - \int 1 \, dx$$ 4. The integral of $$\sec^2(5x)$$ is $$\frac{1}{5} \tan(5x)$$ because the derivative of $$\tan(5x)$$ is $$5 \sec^2(5x)$$. The integral of 1 with respect to $$x$$ is $$x$$. 5. So, the integral becomes: $$\int \tan^2(5x) \, dx = \frac{1}{5} \tan(5x) - x + C$$ 6. Therefore, the final answer is: $$\boxed{\int \tan^2(5x) \, dx = \frac{1}{5} \tan(5x) - x + C}$$