Q1: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be given by $f(x) = \frac{1}{1+x^2}$. Does the improper Riemann integral $$ \int_{-\infty}^{\infty} f(x) , dx$$ exist?

A1: Yes!

A2: No!

A3: One needs more information.

Q2: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be given by $f(x) = \frac{1}{1+x^2}$. Let’s calculate the integral $$ \int_{-\infty}^{\infty} f(x) , dx$$ like shown in the video. Which isolated singularity do we need for that?

A1: $i$

A2: $-i$

A3: $1$

A4: $-1$

A5: $0$

Q3: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be given by $f(x) = \frac{1}{1+x^2}$. Let’s calculate the integral $$ \int_{-\infty}^{\infty} f(x) , dx$$ like shown in the video. What is the value of this integral?

A1: $ 2 \pi i \cdot \frac{1}{2 i} = \pi $

A2: $ 2 \pi i \cdot 0 = 0 $

A3: $ 2 \pi i \cdot \frac{1}{i} = 2 \pi $

A4: $ 2 \pi i \cdot \frac{1}{2} = i \pi $

A5: $ \pi i \cdot \frac{1}{2 i} = \frac{\pi}{2} $