Differentiate ${{\sin }^{-1}}x+x\sqrt{1-{{x}^{2}}}$ with respect to $x$, expressing your answer in i

Question

Differentiate ${{\sin }^{-1}}x+x\sqrt{1-{{x}^{2}}}$ with respect to $x$, expressing your answer in its simplest form. Hence, find $\displaystyle\int{\sqrt{1-{{x}^{2}}}\,\mathrm{d}x}$. [4] Find $\displaystyle\int{\frac{{{x}^{2}}}{\sqrt{4{{x}^{3}}+1}}}\mathrm{d}x$. [2]

Accepted Answer

(a) $\frac{\mathrm d}{\mathrm dx}\left(\sin^{-1}x+x\sqrt{1-x^2}\right)=2\sqrt{1-x^2}$, hence $\displaystyle\int\sqrt{1-x^2}\,\mathrm dx=\frac12\left(\sin^{-1}x+x\sqrt{1-x^2}\right)+C$ (b) $\displaystyle\int\frac{x^2}{\sqrt{4x^3+1}}\,\mathrm dx=\frac16\sqrt{4x^3+1}+C$

2021 NYJC Promo Q1

2021 NYJC Promo Q1

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