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2023 ACJC P1 Q5

2023 ACJC P1 Q5

7 marks
Prelims

  1. Find \(\frac{\mathrm{d}}{\mathrm{d}x}{{\mathrm{e}}^{-{{x}^{2}}}}\). Hence find \(\int x^3 \mathrm{e}^{-x^2} \mathrm{d}x\).

    [3]
  2. By using the substitution \(z={{\mathrm{e}}^{-{{x}^{2}}}}y\), find the general solution of \(\frac{\mathrm{d}y}{\mathrm{d}x}-2xy={{x}^{3}}\), expressing \(y\) in teams of \(x\).[4]

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Answer:(i) \({-2x\mathrm{e}^{-x^2}}\) ; \({-\frac{1}{2}\mathrm{e}^{-x}\left(x^2+1\right)+c}\) (ii) \({y=-\frac{1}{2}\left(x^2+1\right)+c\mathrm{e}^{x^2}}\)

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