Standard +0.3 This is a straightforward area calculation requiring integration of 1/(2x+3) using substitution u=2x+3, then subtracting a rectangular area. The substitution is standard, the limits are simple, and the final manipulation to get the required form involves basic arithmetic. Slightly above average due to the need to recognize the region geometry and handle logarithms, but still a routine P2 question.
3
\includegraphics[max width=\textwidth, alt={}, center]{4ce3208e-8ceb-4848-a9c7-fcda166319f4-04_458_892_269_614}
The diagram shows part of the curve \(y = \frac { 6 } { 2 x + 3 }\). The shaded region is bounded by the curve and the lines \(x = 6\) and \(y = 2\).
Find the exact area of the shaded region, giving your answer in the form \(a - \ln b\), where \(a\) and \(b\) are integers.
3\\
\includegraphics[max width=\textwidth, alt={}, center]{4ce3208e-8ceb-4848-a9c7-fcda166319f4-04_458_892_269_614}
The diagram shows part of the curve $y = \frac { 6 } { 2 x + 3 }$. The shaded region is bounded by the curve and the lines $x = 6$ and $y = 2$.
Find the exact area of the shaded region, giving your answer in the form $a - \ln b$, where $a$ and $b$ are integers.\\
\hfill \mbox{\textit{CAIE P2 2023 Q3 [5]}}