Use the formula ∫ 1 𝑥 𝑑𝑥 = ln(𝑥) + 𝑐, for 𝑥 > 0.

A student is evaluating the indefinite integral $\int \left( 3 + \frac{5}{x} \right) dx$ where $x > 0$. Which of the following is the correct result?

A pharmaceutical company models the rate at which a medication is eliminated from the bloodstream by \(E'(t) = \frac{80}{t+2}\), where \(t\) is the time in hours since administration and \(E'(t)\) is measured in milligrams per hour. At the time of administration (\(t = 0\)), the amount of medication eliminated is \(0\) mg. (a) Use the given rate of elimination to determine an expression for \(E(t)\), the total amount of medication eliminated after \(t\) hours. (2 marks) (b) A second medication has elimination rate \(R'(t) = 45e^{0.15t}\) milligrams per hour. At \(t = 0\), \(R(0) = 0\) mg. Determine the time at which both medications will have eliminated the same total amount from the bloodstream. Give your answer correct to two decimal places. (3 marks)

Determine \(\displaystyle \int \frac{3}{x} \, dx\) for \(x > 0\).