2025 SST PRELIMS P2 Q8

1. Given that \( \int_1^6 f(x)\, dx = 14 \) and \( \int_1^3 f(x)\, dx = 8 \), find
   1. \( \int_3^6 f(x)\, dx \),[1]
   2. \( \int_1^6 (4x - 3f(x))\, dx \).[2]
2. It is given that \( f(x) = \ln(\sin 2x) \). Show that \( f''(x) + [f'(x)]^2 + 4 = 0 \).[5]

Part (b) is independent of part (a): its \(f\) denotes a different function. Prove its identity on intervals where \(\sin2x>0\), so that the real logarithm is defined.