Find $\displaystyle\int{n{{\cos }^{-1}}\left( nx \right)}\mathrm{ d}x$, where $n$ is a positive cons

Question

Find $\displaystyle\int{n{{\cos }^{-1}}\left( nx \right)}\mathrm{ d}x$, where $n$ is a positive constant. [3] Hence find the exact value of $\displaystyle\int_{0}^{\frac{1}{2n}}{n{{\cos }^{-1}}\left( nx \right)\mathrm{ d}x}$. [2]

Accepted Answer

(i) $nx\cos^{-1}(nx)-\sqrt{1-n^2x^2}+C$; (ii) $\frac\pi6-\frac{\sqrt3}{2}+1$

2017 IJC P1 Q2

2017 IJC P1 Q2

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