To solve the integral

$\int \frac{x}{\sec x + 1} \, dx$

we can proceed by using trigonometric identities and substitution.

Step 1: Rewrite the Integral

Rewrite $\sec x$ as $\frac{1}{\cos x}$: $\int \frac{x}{\frac{1}{\cos x} + 1} \, dx = \int \frac{x \cos x}{1 + \cos x} \, dx.$

Step 2: Simplify the Denominator

Observe that $1 + \cos x = 2 \cos^2 \left( \frac{x}{2} \right)$ using the half-angle identity: $1 + \cos x = 2 \cos^2 \left( \frac{x}{2} \right).$ Thus, the integral becomes: $\int \frac{x \cos x}{2 \cos^2 \left( \frac{x}{2} \right)} \, dx = \frac{1}{2} \int \frac{x \cos x}{\cos^2 \left( \frac{x}{2} \right)} \, dx.$

Step 3: Simplify Further

Rewrite $\cos x$ in terms of half-angle formulas if necessary, or consider a substitution to simplify the expression. At this stage, however, a straightforward solution does not emerge directly without further simplification or alternative approaches, such as using series expansion for integration or checking computational resources for an analytical solution.

The integral is non-trivial and may require advanced techniques. Please let me know if you'd like a more detailed analysis or alternative methods such as numerical approaches for an approximate solution.

Would you like additional details or specific explanations?

Here are five related questions:

1. How does the half-angle formula simplify integration problems?
2. What substitution methods are commonly used in trigonometric integrals?
3. Can integrals with $\sec x$ functions be solved with partial fractions?
4. How can we use series expansion for integrals involving trigonometric expressions?
5. What are common techniques for solving non-standard integrals like this?

Tip: When you encounter complex trigonometric integrals, always consider using trigonometric identities or substitutions to simplify the integrand.