2024 DHS H1 Q3

Differentiate \(\ln \left( \frac{\mathrm{e}^{2x}}{\mathrm{e}^{x^2} + 1} \right)\) with respect to \(x\).[3]

Find \(\displaystyle\int (2 - \sqrt{x-3})^2 \, \mathrm{d}x\).[3]

It is given that \(\displaystyle\int_1^6 \mathrm{f}(x) \, \mathrm{d}x = 16\) and \(\displaystyle\int_4^6 \mathrm{f}(x) \, \mathrm{d}x = 7\). Find \(\displaystyle\int_1^4 [5 - 3\mathrm{f}(x)] \, \mathrm{d}x\).[3]