2025 ASRJC P2 Q5
2025 ASRJC P2 Q5
Junior College 2
10 marks
- Solve the following integral.
- \(\displaystyle \displaystyle\int \frac{1-2x}{\sqrt{3+2x-x^2}} \, \mathrm{d}x\).[2]
- \(\displaystyle \displaystyle\int x \ln(2-x^2) \, \mathrm{d}x\), where \(-\sqrt{2} < x < \ \sqrt{2}\).[3]
- A function \(\mathrm{f}\) is defined by \(\mathrm{f}(x) = \mathrm{e}^x - 2\).
- Sketch the graph of \(y = |\mathrm{f}(x)|\), indicating clearly the equation(s) of asymptote(s), if any.[2]
- Hence find \(\displaystyle \displaystyle\int_0^{\ln 3} |\mathrm{f}(x)| \, \mathrm{d}x\), giving your answer in exact form.[3]
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Answer:(a)(i) \({2\sqrt{3 + 2x - x^{2}} - \sin^{-1}\left( \frac{x - 1}{2} \right) + c}\) (ii) \({\frac{x^{2}}{2}\ln(2 - x^{2}) - \frac{x^{2}}{2} - \ln(2 - x^{2}) + c}\) \text{ or } \({\left( \frac{x^{2}}{2} - 1 \right)\ln(2 - x^{2}) - \frac{x^{2}}{2} + c}\) \text{ or } \({\frac{x^{2}}{2}\left( \ln(2 - x^{2}) - 1 \right) - \ln(2 - x^{2}) + c}\) (b)(ii) \({4\ln 2 - 2\ln 3}\) Alternative answers: \({\ln \frac{16}{9};\quad 2\ln \frac{4}{3}}\)
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