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Today, I am going to add a few corollaries (without proof) to the Theorem 2.1 presented in 2 Interesting Integral II.

First, let’s recap the theorem from that blog post:

**Theorem 2.1: **Let [math]f(x)[/math] be an even function and [math]a, b \in \mathbb{R}_{\geq 0}[/math], then

**Theorem 2.1: **Let [math]f(x)[/math] be an even function and [math]a, b \in \mathbb{R}_{\geq 0}[/math], then

[math]\displaystyle \int_{-a}^a \frac{f(x)}{1 + b^x}dx = \frac{1}{2} \int_{-a}^a f(x) dx \tag{1}[/math]

**Proof:**

**Theorem 1.1:** Prove the following

[math]\displaystyle \int_{0}^{\infty} \frac{1}{1 + x^2} \frac{1}{1 + x^{\alpha}}dx = \frac{\pi}{4} \tag{1}[/math]

**Proof:**