Q1: Which of the following functions is a rational function?

A1: $f(x) = \frac{\sin(x)}{x}$

A2: $f(x) = \frac{x^2}{x-\exp(x)}$

A3: $f(x) = \frac{x}{1+x^2}$

A4: $f(x) = \frac{-1}{\tan(x)}$

Q2: What is not an antiderivative of the function $f: \mathbb{R}\setminus {0} \rightarrow \mathbb{R}$ given by $f(x) = \frac{1}{x}$?

A1: $$F(x) = \log(|x|)$$

A2: $$F(x) = \log(|x|) + 2$$

A3: $$F(x) = \begin{cases} \log(|x|) + 2 ,,& x > 0 \ \log(|x|) - 1 ,,& x < 0 \end{cases}$$

A4: $$ F(x) = \log(x) $$

Q3: What is the integral $\int_4^5 \frac{1}{x(x+1)}, dx$?

A1: $\log(5) - \log(6) - \log(4) + \log(5)$

A2: $- \log \left( \frac{25}{24} \right)$

A3: $\log(5) - \log(4)$

A4: $0.04$

A5: $e$