Q1: Let $f: [0,2]\setminus{1} \rightarrow [0,1]$ be an unbounded function. What is the correct definition of the improper Riemann integral $$ \int_0^2 f(x), dx$$

A1: $$ \lim_{\varepsilon \searrow 0 } \int_0^{ 1 - \varepsilon} f(x), dx + \lim_{\varepsilon \searrow 0 } \int_{ 1 + \varepsilon}^{2} f(x), dx $$

A2: $$ \lim_{\varepsilon \searrow 0 } \int_0^{ 2 - \varepsilon} f(x), dx + \lim_{\varepsilon \searrow 0 } \int_{ 1 + \varepsilon}^{2} f(x), dx $$

A3: $$ \lim_{\varepsilon \searrow 0 } \int_0^{ 2 - \varepsilon} f(x), dx + \lim_{\varepsilon \searrow 0 } \int_{ 2 + \varepsilon}^{2} f(x), dx $$

A4: $$ \lim_{\varepsilon \searrow 0 } \int_0^{ 2 - \varepsilon} f(x), dx + \lim_{\varepsilon \searrow 0 } \int_{\varepsilon}^{2} f(x), dx $$

Q2: $f: [-1,1] \setminus {0} \rightarrow [0,1]$ be given by $ f(x) = \frac{1}{x}$. Does the improper Riemann integral exist?

A1: Yes!

A2: No!

Q3: $f: [-1,1] \setminus {0} \rightarrow [0,1]$ be given by $ f(x) = \frac{1}{x}$. Does the Cauchy principal value exist?

A1: Yes!

A2: No!

Q4: $f: [-1,1] \setminus {0} \rightarrow [0,1]$ be given by $ f(x) = \frac{1}{x^3}$. What is the Cauchy principal value $$\mathrm{p.v.} \int_{-1}^1 \frac{1}{x^3} , dx $$ of this function?

A1: $0$

A2: $1$

A3: $\frac{1}{2}$

A4: $\frac{1}{3}$

A5: $3$