https://file.aiquickdraw.com/mathbot/file/1726535006402naphr42r.jpg

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.

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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 \( x \).

Consider the function g(x) defined by g(x) = ∫(0 to π) [sin(x + t) / (1 + cos²(t))] dt.

Explore the integral function g(x) = ∫(0 to π) [sin(x + t) / (1 + cos²(t))] dt, which combines trigonometric functions and advanced calculus concepts. Learn how to apply Leibniz's Rule for differentiation under the integral sign and investigate its behavior at various values of x. Suitable for advanced calculus students.

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