2023 VJC P1 Q10
2023 VJC P1 Q10
11 marks
Prelims
- Find \(\displaystyle\displaystyle \displaystyle\int{\frac{3-2x}{\sqrt{8+2x-{{x}^{2}}}}}\ \mathrm{d}x\).[4]
- Find the exact value of \(\displaystyle\displaystyle \displaystyle\int_{-\frac{\pi }{8}}^{\frac{\pi }{6}}{\sec }2x\left| \tan 2x \right|\ \mathrm{d}x\).[3]
- Find the exact value of \(\displaystyle\displaystyle \displaystyle\int_{0}^{\frac{\pi }{3}}{\frac{1}{1+8{{\cos }^{2}}x}}\ \mathrm{d}x\) using the substitution \(t=\tan x\).[4]
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Answer:(a) \(2\sqrt{8+2x-x^2}+\sin^{-1}\!\left(\dfrac{x-1}{3}\right)+C\) (b) \(\dfrac{\sqrt2}{2}\) (c) \(\dfrac\pi{18}\)
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