When an integrand is a sum of terms, especially after simplification, always check if it matches the form of a product rule derivative, $u'v+uv'$. This pattern recognition can turn a potentially difficult integral into a trivial one.
Step 1: Simplify the integrand
We know \( 1 + \tan^2 x = \sec^2 x \).
So the integral is \( I = \int \sec^2 x (1 + 2x \tan x) dx \).
\( I = \int (\sec^2 x + 2x \sec^2 x \tan x) dx \).
Step 2: Identify the derivative form
Check the derivative of \( x \sec^2 x \):
\[ \frac{d}{dx} (x \sec^2 x) = 1 \cdot \sec^2 x + x \cdot \frac{d}{dx}(\sec^2 x) \]
\[ = \sec^2 x + x \cdot (2 \sec x \cdot \sec x \tan x) \]
\[ = \sec^2 x + 2x \sec^2 x \tan x \]
\[ = \sec^2 x (1 + 2x \tan x) \]
Step 3: Conclusion
Since the integrand is exactly the derivative of \( x \sec^2 x \), the integral is:
\[ I = x \sec^2 x + c \]