2025 HCI Promo Q3

2025 HCI Promo Q3

Junior College 1
7 marks
    1. Write down \(\dfrac{d}{dx}\mathrm{e}^{\cos 2x}\).[1]
    2. Hence find \(\int (\sin 2x)\mathrm{e}^{\cos 2x+\mathrm{e}^{\cos 2x}}\, dx\).[2]
  2. Use the substitution \(x=\sqrt{3}\sec\theta\) to find \(\int \dfrac{1}{x^2\sqrt{x^2-3}}\, dx\) where \(0\leq \theta < \dfrac{\pi}{2}\).[4]

Solution:

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Answer:(a)(i) \(-2\sin2x\,e^{\cos2x}\) (a)(ii) \(-\dfrac12e^{e^{\cos2x}}+C\) (b) \(\dfrac{\sqrt{x^2-3}}{3x}+C\)

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