2025 YIJC Promo Q10

2025 YIJC Promo Q10

Junior College 1
12 marks
  1. Find \(\displaystyle \int \frac{2}{5-4x^2}\, dx\).[2]
  2. Find \(\displaystyle \int 3x\left(x^2-1\right)^8\, dx\).[2]
  3. Find \(\displaystyle \int \ln\left(x^2+3\right)\, dx\).[4]
  4. Use the substitution \(x=\sin\theta\) to find the exact value of \(\displaystyle \int_0^1 \sqrt{1-x^2}\, dx\).[4]

Solution:

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Answer:\(\dfrac{\sqrt5}{10}\ln\left|\dfrac{\sqrt5+2x}{\sqrt5-2x}\right|+C\); \(\dfrac16(x^2-1)^9+C\); \(x\ln(x^2+3)-2x+2\sqrt3\tan^{-1}(x/\sqrt3)+C\); \(\pi/4\)

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