Find \(\displaystyle\displaystyle \displaystyle\int{\frac{3-2x}{\sqrt{5+4x-{{x}^{2}}}}}\,\mathrm{d}x

Question

Find $\displaystyle\displaystyle \displaystyle\int{\frac{3-2x}{\sqrt{5+4x-{{x}^{2}}}}}\,\mathrm{d}x$.[4] Show that $\mathop{\displaystyle\displaystyle \displaystyle\int }_{\sqrt{2}}^{4}\left| x-3 \right|\mathrm{d}xx=a-b\sqrt{2}$, where $a$ and $b$ are constants to be determined.[3] Using the substitution ${x=\sin u}$, find the exact value of ${\int_{\frac{\sqrt{2}}{2}}^{\frac{\sqrt{3}}{2}} \frac{1+x^2}{\sqrt{1-x^2}} \mathrm{d}x}$. [4]

Accepted Answer

(a) $2\sqrt{5+4x-{{x}^{2}}}-{{\sin }^{-1}}(\frac{x-2}{3})+C$ (b) $a=6$, $b=3$ (c) $\frac{\pi }{8}-\frac{\sqrt{3}}{8}+\frac{1}{4}$

2023 VJC Promo Q6

2023 VJC Promo Q6

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