2020 ACJC P1 Q1
2020 ACJC P1 Q1
6 marks
Prelims
- Use the substitution \(u=2x-1\) to find \(\displaystyle\displaystyle \displaystyle\int{\frac{x}{\sqrt{1-{{\left( 2x-1 \right)}^{2}}}}\mathrm{d}x}\).[4]
- Hence find \(\displaystyle\displaystyle \displaystyle\int{{{\sin }^{-1}}\left( 2x-1 \right)\mathrm{d}x}\).[2]
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Answer:(i) \(-\frac{1}{4}\sqrt{1-(2x-1)^2} + \frac{1}{4}\sin^{-1}(2x-1) + C\) (ii) \(\left(x - \frac{1}{2}\right)\sin^{-1}(2x-1) + \frac{1}{2}\sqrt{1-(2x-1)^2} + C\)
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