2022 NYJC Promo Q2
2022 NYJC Promo Q2
6 marks
Promo
- Show that \(\frac{4{{x}^{2}}-6x+1}{\left( {{x}^{2}}+1 \right)\left( x-2 \right)}\) can be expressed as \(\frac{Ax+B}{{{x}^{2}}+1}-\frac{C}{x-2}\), where \(A\), \(B\) and \(C\) are constants to be determined. Hence find \(\displaystyle \displaystyle\int{\frac{4{{x}^{2}}-6x+1}{\left( {{x}^{2}}+1 \right)\left( x-2 \right)}\mathrm{d}x}\).[3]
- Differentiate \({{\mathrm{e}}^{\sqrt{x}}}\) with respect to \(x\). Hence, or otherwise, find \(\displaystyle \displaystyle\int{\frac{{{\mathrm{e}}^{\sqrt{x}}}}{\sqrt{x}{{\left( {{e}^{\sqrt{x}}}+1 \right)}^{4}}}\mathrm{d}x}\).[3]
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Answer:(a) \(A=3, B=0, C=-1\); \(\frac32\ln(x^2+1)+\ln|x-2|+K\). (b) \(\frac{\mathrm d}{\mathrm dx}e^{\sqrt{x}}=\frac{e^{\sqrt{x}}}{2\sqrt{x}}\); \(-\frac23(e^{\sqrt{x}}+1)^{-3}+K\).
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