The function provided in the image is defined as follows:

$g(x) = \int_0^{\pi} \frac{\sin(x + t)}{1 + \cos^2(t)} \, dt$

This is an integral function $g(x)$ where the integrand involves a combination of sine and cosine terms. Specifically, the sine term is a function of both $x$ and $t$, and the denominator is a sum involving $\cos^2(t)$.

Would you like to:

• Compute this integral directly?
• Differentiate $g(x)$ using the Leibniz rule?
• Investigate the behavior of the function for specific values of $x$?
• Analyze properties like symmetry or periodicity?

Let me know how you'd like to proceed, or I can provide a detailed solution directly.

Here are some related questions you might find interesting:

1. What is the derivative of $g(x)$ with respect to $x$?
2. How does $g(x)$ behave as $x \to 0$?
3. Can the integral be computed exactly, or does it require numerical methods?
4. How does the function $g(x)$ relate to the Fourier transform?
5. What are the symmetry properties of the integrand?

Tip: Using Leibniz’s rule is often helpful when differentiating under the integral sign with respect to a parameter like $x$.