The given integral is:

$I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( \frac{\cos x}{\pi - x + 1} \right) \pi^{-x} \, dx$

Approach to Solve the Integral:

We need to evaluate this definite integral over the symmetric interval $[- \frac{\pi}{2}, \frac{\pi}{2}]$. Here are a few key points to consider for simplifying or solving the integral:

1. Odd and Even Function Symmetry: The integral is defined over a symmetric interval, so we can check if the integrand is odd or even. If the integrand is odd, the integral will be zero. If it is even, we can simplify by doubling the integral over $[0, \frac{\pi}{2}]$.

   • $\cos x$ is an even function: $\cos(-x) = \cos(x)$.
   • $\pi^{-x}$ is neither odd nor even but is symmetric around zero: $\pi^{-(-x)} = \pi^x$.
   • The term $\frac{1}{\pi - x + 1}$ becomes $\frac{1}{\pi + x + 1}$ when $x$ is replaced by $-x$, which changes slightly.

   So, the integrand does not behave as an odd or even function, and we cannot directly simplify using symmetry.

2. Substitution: We could try a substitution like $u = \pi - x$ or use numerical methods, as the function is not straightforward to integrate using elementary techniques.

Let me know if you would like a numerical approximation or further steps in the analytical approach!

Do you have any questions or would like further clarification?

Here are 5 questions you might find helpful:

1. What are the properties of odd and even functions in definite integrals over symmetric intervals?
2. How can substitution methods help in simplifying complex integrals?
3. When is numerical integration preferred over symbolic methods?
4. How does changing the limits of integration affect the outcome of definite integrals?
5. What are some useful strategies for handling integrals with non-standard functions like $\pi^{-x}$?

Tip:

For integrals over symmetric intervals, always check if the integrand is odd or even. This can sometimes simplify the problem greatly or even give a result immediately.