The problem asks to show the identity:

$\Gamma(n) \Gamma(1 - n) = \frac{\pi}{\sin(n \pi)}$

Given the integral:

$\int_0^{\infty} \frac{x^{n-1}}{1+x} dx = \frac{\pi}{\sin(n\pi)}$

This result is closely related to the Euler reflection formula for the Gamma function, which is:

$\Gamma(n)\Gamma(1 - n) = \frac{\pi}{\sin(\pi n)}$

Proof outline:

1. Use the provided integral: The integral $\int_0^{\infty} \frac{x^{n-1}}{1+x} dx$ is a well-known integral related to the Beta function $B(x,y)$ and can be expressed in terms of Gamma functions using the relationship: $B(n, 1 - n) = \int_0^{\infty} \frac{x^{n-1}}{1+x} dx$ where $B(n, 1-n) = \frac{\Gamma(n)\Gamma(1-n)}{\Gamma(1)}$. Since $\Gamma(1) = 1$, we have: $B(n, 1 - n) = \Gamma(n)\Gamma(1 - n)$

2. Relate the Beta function to the given integral: From the given information, we already know that the integral evaluates to $\frac{\pi}{\sin(n\pi)}$, hence: $\Gamma(n)\Gamma(1 - n) = \frac{\pi}{\sin(n\pi)}$

This completes the proof. Therefore, the identity $\Gamma(n) \Gamma(1 - n) = \frac{\pi}{\sin(n\pi)}$ is verified using the given integral.

Would you like more details or a deeper exploration into the steps of this proof?

Here are 5 related questions to expand on this concept:

1. What is the Euler reflection formula and where is it used in other mathematical contexts?
2. How is the Beta function related to the Gamma function, and how does it arise in integrals?
3. Can you prove the integral formula $\int_0^{\infty} \frac{x^{n-1}}{1+x} dx = \frac{\pi}{\sin(n\pi)}$ independently?
4. How does the Gamma function generalize the factorial function for non-integer values?
5. What are some common applications of the Gamma function in physics and engineering?

Tip: When dealing with Gamma functions and integrals, recognizing common forms like the Beta function can simplify proofs and connections between different areas of mathematics.