2024 ACS(BR) P1 Q9

2024 ACS(BR) P1 Q9

Secondary 4
9 marks
  1. Differentiate \(x\sin x+\cos x\) with respect to \(x\).[2]
  2. Show that \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac13x\sin^3x\right)=\dfrac13\sin^3x-x\cos^3x+x\cos x\).[3]
  3. Using the results found in part (a) and part (b), find \(\displaystyle\int(\sin^3x-3x\cos^3x)\,\mathrm{d}x\).[4]

Solution:

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Answer:(a) \(x\cos x\). (b) \(\dfrac13\sin^3x-x\cos^3x+x\cos x\). (c) \(x\sin^3x-3x\sin x-3\cos x+C\).

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